#6677303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂×#3119	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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)(3,4)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22)(23,30)(24,35)(25,33)(26,29)(27,28)(31,32)(34,36);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22,26,29)(23,31,33,36)(24,28,27,35)(25,32,30,34);(2,4)(21,23)(22,25)(24,34)(26,33)(27,32)(28,36)(29,30)(31,35);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,26,27)(22,28,29,35)(23,32,33,34)(25,31,30,36);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25)(22,23)(24,31)(26,30)(27,36)(28,32)(29,33)(34,35)

Polycyclic presentation:

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

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, 6584499);