#195696

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102892);