#474336

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efghi,c^2=fi,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381532);