#381109

order	512
nilpotency class	3
isoclinism	isoclinic class of #195981
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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1170 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)(17,18,23,27,22,26,21,25)(19,24,28,31,20,29,30,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,22)(21,30)(23,28)(25,32)(26,29)(27,31);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,22,19)(18,24,26,29)(21,30,23,28)(25,31,27,32);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,27)(19,28,20,30)(24,31,29,32)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288305);