#364037

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271233);