#105418

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,g^2=h,b^a=bd,d^a=dh,f^a=fhi,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12614);