#382786

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
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

1266 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)(11,14)(12,13)(15,16)(17,18)(19,22)(20,26)(21,27)(23,25)(24,29)(28,31)(30,32);(2,5)(6,8)(17,19,22,18)(20,24,29,26)(21,27,23,25)(28,31,30,32);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,23,28,22,29,21,30)(18,24,25,32,19,26,27,31);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,19,27)(20,28,29,30)(24,31,26,32)

Polycyclic presentation:

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

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, 289982);