#356268

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1726 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=ei,b^a=be,c^a=cf,f^a=fi,d^b=de,g^c=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263464);