#356924

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1074 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,28)(25,27)(29,31)(30,33)(32,34)(35,36);(6,9)(7,11)(13,15)(14,17)(16,20)(18,19)(21,23,24,30,25,31,32,36)(22,26,28,33,27,29,34,35);(1,2,4,3)(14,18)(17,19)(21,22)(23,29)(24,28)(25,27)(26,31)(30,35)(32,34)(33,36);(1,3,4,2)(5,7,8,11)(6,10,9,12)(15,20)(17,19)(22,27)(26,29)(28,34)(33,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264120);