#356576

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1600 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=fg,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,e^d=ei,h^d=hi

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, 263772);