#6629507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2258 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)(13,14,20,26)(15,22,29,35)(16,21,28,36)(17,23,31,38)(18,25,19,24)(27,37,41,44)(30,34,42,43)(32,40,33,39);(6,9)(10,12)(13,15,18,16,20,29,19,28)(14,21,24,22,26,36,25,35)(17,27,32,30,31,41,33,42)(23,34,39,37,38,43,40,44);(5,7,8,11)(6,10,9,12)(13,16,18,29,20,28,19,15)(14,22,24,36,26,35,25,21)(17,30,32,41,31,42,33,27)(23,37,39,43,38,44,40,34);(2,4)(13,17)(14,23)(15,27)(16,30)(18,32)(19,33)(20,31)(21,34)(22,37)(24,39)(25,40)(26,38)(28,42)(29,41)(35,44)(36,43);(17,31)(23,38)(27,41)(30,42)(32,33)(34,43)(37,44)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536703);