#355817

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1336 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cei,d^b=di,f^c=fi

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, 263013);