#465328

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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

1092 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:

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

Polycyclic presentation:

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

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, 372524);