#367630

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi

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, 274826);