#362946

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1306 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi

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, 270142);