#6520545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#25793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei

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, 6427741);