#196053

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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,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, 103249);