#387091

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,24,22,27,19,28)(20,26,21,25,30,31,29,32);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,27,28)(20,29,30,21)(25,31,32,26);(1,4,5,8)(2,6,3,7)(11,16)(13,15)(17,20,22,30)(18,25,27,32)(19,29,23,21)(24,31,28,26);(2,3)(6,7)(10,14)(11,16)(17,21)(18,26)(19,20)(22,29)(23,30)(24,25)(27,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^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, 294287);