#196685

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
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	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1354 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)(11,17)(15,20)(16,19)(21,22,25,30,26,29,24,28)(23,31,32,27,34,35,33,36);(5,7,8,16,10,15,9,17)(6,11,18,20,13,19,12,14)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,36)(30,31);(2,4)(7,15)(11,19)(14,20)(16,17)(21,24,26,25)(22,28,29,30)(23,32,34,33)(27,35,36,31);(5,8,10,9)(6,12,13,18)(7,16,15,17)(11,14,19,20)(22,29)(27,36)(28,30)(31,35)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=h,e^2=hi,g^2=i,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, 103881);