#6816879

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	C₂×#547	C₂²
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) = 256

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ghi,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

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, 6724075);