#222901

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1210 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dei,g^c=gh

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, 130097);