#393730

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, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dg,c^b=ci,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, 300926);