#384729

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,22,26,29,25,27,24,33)(23,30,31,35,32,28,34,36);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(13,15,17,16)(14,18,19,20)(22,27)(28,30)(29,33)(35,36);(2,4)(5,8,10,9)(6,11,7,12)(13,16,17,15)(14,18,19,20)(21,23)(22,28)(24,34)(25,32)(26,31)(27,30)(29,36)(33,35);(5,9,10,8)(6,11,7,12)(14,19)(18,20)(21,24,25,26)(22,29,27,33)(23,31,32,34)(28,35,30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=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, 291925);