#6685112

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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,32)(24,31)(25,33)(26,30)(27,34)(28,36)(29,35)(37,42)(38,41)(39,43)(40,44);(6,9)(7,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,33)(21,31)(22,32)(26,29)(27,39)(28,37)(30,35)(34,43)(36,42)(38,40)(41,44);(1,2,3,4)(14,20)(15,23)(16,27)(18,29)(19,30)(24,38)(25,26)(31,42)(32,33)(36,44)(37,39)(41,43);(6,9)(10,12)(13,16,17,27)(14,21,20,34)(15,24,25,39)(18,28,29,40)(19,31,33,43)(22,36,35,44)(23,37,26,38)(30,41,32,42);(5,7,8,11)(6,10,9,12)(15,23)(18,29)(19,32)(22,35)(24,37)(25,26)(28,40)(30,33)(31,42)(36,44)(38,39)(41,43)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6592308);