#6630347

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
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	#32939	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,26)(17,22)(18,32)(19,23)(21,27)(24,36)(25,33)(28,40)(29,34)(30,37)(31,38)(35,41)(39,44)(42,43);(6,9)(10,12)(13,15,19,18)(14,20,23,32)(16,27,29,24)(17,30,31,25)(21,34,36,26)(22,37,38,33)(28,39,42,41)(35,40,44,43);(13,16,18,27,19,29,15,24)(14,21,20,26,23,36,32,34)(17,28,30,41,31,42,25,39)(22,35,33,40,38,44,37,43);(2,4)(13,17)(14,22)(15,25)(16,28)(18,30)(19,31)(20,33)(21,35)(23,38)(24,39)(26,40)(27,41)(29,42)(32,37)(34,43)(36,44);(5,7,8,11)(6,10,9,12)(16,29)(17,31)(21,36)(22,38)(24,27)(25,30)(26,34)(33,37)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ci,d^b=di,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537543);