#453770

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

630 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:

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

Polycyclic presentation:

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

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, 360966);