#495215

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320128, #72323051
maximal normal subgroup of	#72320128, #72323051
maximal quotient of	#72320212

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402411);