#391340

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298536);