#385370

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1278 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=efi,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292566);