#363018

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

1098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270214);