#196001

order	512
nilpotency class	3
isoclinism	isoclinic class of #195990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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 ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1441 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)(7,11,17,20)(8,12)(9,13)(10,14)(15,18,16,19)(21,22,25,30,26,29,24,28)(23,27,31,34,32,35,33,36);(5,7)(6,11,14,20)(8,15)(9,16)(10,17)(12,18,13,19)(21,23,25,33,26,32,24,31)(22,27,28,34,29,35,30,36);(2,4)(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,24,26,25)(22,28,29,30)(23,31,32,33)(27,34,35,36);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(23,32)(27,35)(31,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103197);