#208534

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂²×#121	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1260 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)(11,13,15,16)(12,14)(17,18,22,25,21,24,30,19)(20,27,29,26,28,23,31,32);(9,11)(10,13,14,16)(12,15)(17,19,22,24,21,25,30,18)(20,26,29,27,28,32,31,23);(2,5)(6,8)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31);(1,3,4,7)(2,6,5,8)(20,28)(23,27)(26,32)(29,31)

Polycyclic presentation:

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

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, 115730);