#219670

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19710	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

496 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,6)(4,15,19,13)(5,20)(7,11)(8,12)(9,18,25,17)(10,21)(14,23)(16,29,32,24)(22,26)(27,30,28,31)(33,34,39,43,37,42,38,54)(35,44,48,59,46,40,47,56)(36,49,53,58,51,63,52,41)(45,60,62,55,50,64,61,57);(1,3,7,12,8,11,6,2)(4,9,17,15,19,25,18,13)(5,20,22,10,23,14,21,26)(16,29,30,27,32,24,31,28)(33,35,38,47,37,46,39,48)(34,40,54,59,42,44,43,56)(36,50,52,62,51,45,53,61)(41,57,63,64,58,55,49,60);(1,4,8,19)(2,9)(3,13)(5,16,23,32)(6,17,7,18)(10,24)(11,15)(12,25)(14,27)(20,28)(21,30,22,31)(26,29)(33,36,39,53,37,51,38,52)(34,41,54,63,42,58,43,49)(35,45,48,62,46,50,47,61)(40,55,59,60,44,57,56,64);(1,5,6,21,8,23,7,22)(2,10,11,20,12,26,3,14)(4,16,17,30,19,32,18,31)(9,24,15,28,25,29,13,27)(34,42)(36,51)(40,44)(43,54)(45,50)(52,53)(56,59)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efh,d^2=ei,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126866);