#219611

order	512
nilpotency class	3
isoclinism	isoclinic class of #219600
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	#19661	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,5)(4,11)(6,10)(7,13)(8,14)(9,15)(12,16)(17,18)(19,22)(20,31,35,29)(21,36)(23,27)(24,28)(25,34,41,33)(26,37)(30,39)(32,45,48,40)(38,42)(43,46,44,47);(1,3,5,2)(4,7,12,9)(6,10,14,8)(11,15,16,13)(17,19,23,28,24,27,22,18)(20,25,33,31,35,41,34,29)(21,30,38,42,39,36,37,26)(32,40,46,44,48,45,47,43);(1,4,6,13,5,12,14,15)(2,7,8,11,3,9,10,16)(17,20)(18,25,28,41)(19,29,27,31)(21,32)(22,33)(23,34)(24,35)(26,40,42,45)(30,43,36,44)(37,46)(38,47)(39,48);(2,3)(7,9)(8,10)(11,16)(17,21,23,38,24,39,22,37)(18,26,19,30,28,42,27,36)(20,32,34,47,35,48,33,46)(25,40,29,43,41,45,31,44)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286568, #72286682
maximal normal subgroup of	#72286568, #72286682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126807);