#387992

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=eg,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295188);