#346394

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3493	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253590);