#369471

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

1258 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fgi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276667);