#382285

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=ei,d^2=efhi,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289481);