#6817557

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei

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, 6724753);