#6440809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2744 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:

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

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,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,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, 6348005);