#6682258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2202 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)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14,19,24)(15,25,28,38)(16,23,18,21)(17,30,32,36)(20,33,35,26)(22,31,37,29)(27,34,42,44)(39,40,43,41);(1,2,3,4)(13,15,19,28)(14,20,24,35)(16,26,18,33)(17,27,32,42)(21,25,23,38)(22,34,37,44)(29,40,31,41)(30,39,36,43);(6,10)(9,11)(13,16,19,18)(14,21,24,23)(15,26,28,33)(17,31,32,29)(20,25,35,38)(22,36,37,30)(27,41,42,40)(34,43,44,39);(7,12)(9,11)(13,17,19,32)(14,22,24,37)(15,27,28,42)(16,29,18,31)(20,34,35,44)(21,30,23,36)(25,39,38,43)(26,40,33,41)

Polycyclic presentation:

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

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, 6589454);