#498764

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2397	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

609 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cefi,d^b=de,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405960);