#382264

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

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	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,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, 289460);