#416461

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

772 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^c=gi

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, 323657);