#355629

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

1030 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei,d^b=di,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, 262825);