#371288

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1442 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=gi,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, 278484);