#6357826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357820
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60648	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(3,7)(6,8)(9,10,15,19)(11,20,24,35)(12,17,26,33)(13,18,14,28)(16,22,30,23)(21,34,37,39)(25,31,27,32)(29,38,40,36);(9,11,13,22,15,24,14,23)(10,16,28,35,19,30,18,20)(12,21,27,38,26,37,25,36)(17,29,31,39,33,40,32,34);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,12)(10,17)(11,21)(13,27)(14,25)(15,26)(16,29)(18,32)(19,33)(20,34)(22,38)(23,36)(24,37)(28,31)(30,40)(35,39);(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,25,26,27)(16,20,30,35)(17,31,33,32)(21,36,37,38)(29,39,40,34)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^c=di,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265022);