#464853

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1300 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,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, 372049);