#371542

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,3)(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,25,20,24,29,23)(19,26,28,31,27,22,30,32);(1,2,4,3)(5,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,22)(20,27)(21,28)(23,31)(24,26)(25,32)(29,30);(11,16)(13,15)(18,23)(21,29)(22,31)(24,25)(26,32)(28,30);(1,3,4,2)(10,14)(13,15)(18,24)(19,27)(23,25)(28,30)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,h^c=hi

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, 278738);