#196412

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1467 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)(11,15)(16,20)(17,19)(21,22)(23,26,30,34)(24,27)(25,28)(29,33,31,35)(32,36);(5,7,9,15,10,17,8,16)(6,11,12,14,13,20,18,19)(21,23)(22,26,28,34)(24,31)(25,30)(27,35,36,33)(29,32);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,20,14)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,33,34,35);(5,8,10,9)(6,12,13,18)(7,16,17,15)(11,14,20,19)(22,28)(26,34)(27,36)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103608);