#6456528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35212	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1907 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363724);