#465275

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|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, 372471);