#439556

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

1254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=efi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346752);