#6687723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	#30704	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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)(21,22,26,30)(23,27,33,34)(24,32,35,31)(25,28,36,29);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,29)(25,32)(26,33)(28,35)(30,34)(31,36);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,25,34,26,35,36,27)(22,23,29,32,30,33,28,31);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,25,26,36)(22,28,30,29)(23,31,33,32)(24,34,35,27);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594919);