#451330

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

832 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ef,c^2=e,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358526);