#377574

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1330 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)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,31)(28,32);(10,13)(14,16)(17,19,21,29)(18,23,24,26)(20,28,22,27)(25,32,30,31);(2,4)(9,11,12,15)(10,14,13,16)(18,24)(19,27)(20,22)(23,31)(26,32)(28,29);(5,7,8,6)(10,13)(14,16)(17,20,21,22)(18,25,24,30)(19,28,29,27)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ei,d^2=fh,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284770);