#6558125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2519 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465321);