#6811466

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#39357	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 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)(8,12)(9,11)(13,14)(15,23)(16,27)(17,21)(18,32)(19,25)(20,30)(22,31)(24,40)(26,35)(28,44)(29,38)(33,37)(34,43)(36,41)(39,42);(2,3)(5,7,10,6)(8,12,9,11)(13,15)(14,19)(16,28)(17,25)(18,26)(20,36)(21,23)(22,35)(24,29)(27,40)(30,43)(31,42)(32,39)(33,41)(34,37)(38,44);(1,2,4,3)(13,16,17,29)(14,20,21,37)(15,24,25,28)(18,33,31,30)(19,34,23,36)(22,38,32,27)(26,43,42,41)(35,40,39,44);(5,8,10,9)(6,11,7,12)(14,21)(15,25)(20,37)(22,32)(24,28)(26,42)(27,38)(41,43);(1,3,4,2)(5,9,10,8)(6,11,7,12)(14,21)(19,23)(20,37)(22,32)(27,38)(34,36)(35,39)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718662);