#476741

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383937);