#6617598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24218	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3063 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524794);