#496105

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

780 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,c^2=f,d^2=gi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210570
maximal normal subgroup of	#72210570
maximal quotient of	#72210403, #72210421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403301);