#495026

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

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fhi,f^2=hi,h^2=i,b^a=be,c^a=cf,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, #72323068
maximal normal subgroup of	#72320128, #72323068
maximal quotient of	#72320179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402222);