#6513371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂×#3541	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3048 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)(13,14,18,23,19,24,16,21)(15,25,26,35,28,38,33,20)(17,30,31,36,32,22,29,37)(27,41,40,34,43,39,42,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,35)(16,26,18,33)(17,27,32,43)(21,25,23,38)(22,34,30,44)(29,40,31,42)(36,39,37,41);(2,4)(6,9)(7,11)(13,16,19,18)(14,21,24,23)(15,26,28,33)(17,29,32,31)(20,25,35,38)(22,36,30,37)(27,40,43,42)(34,39,44,41);(5,7,8,11)(6,10,9,12)(13,17,19,32)(14,22,24,30)(15,27,28,43)(16,29,18,31)(20,34,35,44)(21,36,23,37)(25,39,38,41)(26,40,33,42);(6,9)(10,12)(13,16,19,18)(14,21,24,23)(15,26,28,33)(17,31,32,29)(20,25,35,38)(22,37,30,36)(27,42,43,40)(34,41,44,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6420567);