#6545009

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(13,14,17,22,18,21,30,35)(15,19,25,32,24,33,40,44)(16,26,27,34,28,36,29,20)(23,37,39,42,38,43,41,31);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,33)(22,32)(26,37)(27,39)(28,38)(29,41)(30,40)(34,42)(35,44)(36,43);(6,9)(10,12)(13,16,18,28)(14,20,21,34)(15,23,24,38)(17,29,30,27)(19,31,33,42)(22,36,35,26)(25,41,40,39)(32,43,44,37);(5,7)(6,10)(8,11)(9,12)(14,21)(15,24)(20,34)(22,35)(23,38)(25,40)(26,36)(39,41);(7,11)(10,12)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452205);