#196055

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1208 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)(5,6,9,14,10,13,8,12)(7,11,15,18,16,19,17,20)(21,22,26,30,27,23,25,29)(24,28,32,35,33,31,34,36);(5,7,8,15,10,16,9,17)(6,11,14,20,13,19,12,18)(21,23,26,29,27,22,25,30)(24,31,32,36,33,28,34,35);(2,4)(5,8,10,9)(6,12,13,14)(7,15,16,17)(11,18,19,20)(21,24)(22,28)(23,31)(25,34)(26,32)(27,33)(29,36)(30,35);(21,25,27,26)(22,29,23,30)(24,32,33,34)(28,35,31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103251);