#6815678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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	#31037	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,26,29)(23,30,34,27)(24,35,28,25)(31,32,36,33);(2,3)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23)(22,27)(24,36)(25,33)(26,34)(28,31)(29,30)(32,35);(1,2,4,3)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,25,22,26,28,35,29)(23,31,33,30,34,36,32,27);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,26,35)(22,28,29,24)(23,32,34,33)(27,36,30,31);(1,3,4,2)(7,17)(11,14)(15,16)(19,20)(21,25,26,35)(22,24,29,28)(23,33,34,32)(27,36,30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722874);