#196442

order	512
nilpotency class	3
isoclinism	isoclinic class of #161570
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35031	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1515 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22)(23,26,30,33)(24,27)(25,28)(29,32,35,36)(31,34);(5,7,9,16,10,17,8,15)(6,11,12,14,13,19,18,20)(21,23)(22,26,28,33)(24,29)(25,30)(27,32,34,36)(31,35);(2,4)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(24,31)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103638);