#195331

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1134 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102527);