#392392

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,h^d=hi

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, 299588);