#6816919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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

2132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724115);