#381154

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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

1446 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)(3,4)(6,9)(10,12)(13,14)(15,20)(16,18)(17,19)(21,22,26,28,25,30,24,29)(23,27,33,36,31,35,32,34);(5,6,8,9)(7,10,11,12)(13,15,17,16)(14,18,19,20)(21,23)(22,27)(24,32)(25,31)(26,33)(28,36)(29,34)(30,35);(2,4)(5,7)(6,10)(8,11)(9,12)(15,16)(18,20)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(7,11)(10,12)(13,16,17,15)(14,18,19,20)(23,31)(27,35)(32,33)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fi,h^c=hi

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, 288350);