#6584375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2083 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491571);