#372852

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

1398 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280048);