#364133

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=egi,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271329);