#378979

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

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	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1422 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:

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

Polycyclic presentation:

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

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, 286175);