#371604

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

1152 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)(11,14)(12,13)(15,16)(17,18,22,24,21,25,20,26)(19,23,29,32,27,31,28,30);(1,2,4,3)(5,7,8,6)(10,13)(11,15)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,31)(26,30);(1,2,4,3)(9,11,12,15)(10,14,13,16)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(1,3,4,2)(10,13)(14,16)(18,25)(19,27)(24,26)(28,29)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=e,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,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, 278800);