#7196692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#59824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=gh,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi

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, 7103888);