#358373

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

Aliases

No aliases for this group.

Subgroups

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

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=di

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, 265569);