#126805

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fg,c^b=ci,d^b=di,e^b=ei

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, 34001);