#415514

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

688 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=f,c^2=e,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322710);