#393296

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306950, #72307302, #72307320
maximal normal subgroup of	#72306950, #72307302, #72307320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300492);