#355016

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

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×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262212);