#208548

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#392	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1220 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)(4,5)(7,8)(9,10,12,15,13,14,16,11)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(9,11,12,14,13,15,16,10)(17,19)(18,22,24,29)(20,25)(21,26)(23,28,30,32)(27,31);(2,5)(6,8)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,3,4,7)(2,6,5,8)(20,27)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115744);