#215540

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1144 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)(3,7)(4,6)(5,8)(10,13,14,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,3,4,8)(2,5,6,7)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32);(3,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31);(2,6)(5,7)(19,26)(22,30)(25,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122736);