#195753

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
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

1104 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)(5,6,9,11,10,14,8,13)(7,15,16,12,18,19,17,20)(21,22,26,32,27,31,25,30)(23,33,34,28,29,36,35,24);(1,2,3,4)(6,11)(8,9)(12,15)(13,14)(16,17)(19,20)(21,23,27,29)(22,28,31,24)(25,34,26,35)(30,36,32,33);(5,7)(6,12)(8,16)(9,17)(10,18)(11,15)(13,19)(14,20)(21,24,27,28)(22,29,31,23)(25,33,26,36)(30,35,32,34);(5,8,10,9)(6,11,14,13)(7,16,18,17)(12,15,20,19)(21,25,27,26)(22,30,31,32)(23,34,29,35)(24,33,28,36)

Polycyclic presentation:

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

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, 102949);