#6544813

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×#3275	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2052 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)(13,14,19,22,18,23,17,24)(15,20,26,37,27,35,28,34)(16,29,30,21,32,38,31,36)(25,39,42,33,41,43,40,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,28)(18,27)(19,26)(21,33)(22,37)(23,35)(24,34)(29,39)(30,42)(31,40)(32,41)(36,44)(38,43);(6,9)(10,12)(13,16,18,32)(14,21,23,36)(15,25,27,41)(17,30,19,31)(20,33,35,44)(22,29,24,38)(26,40,28,42)(34,43,37,39);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,24)(15,26,27,28)(16,30,32,31)(20,34,35,37)(21,29,36,38)(25,40,41,42)(33,43,44,39);(7,11)(10,12)(14,23)(15,27)(21,36)(22,24)(25,41)(26,28)(29,38)(40,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=ei,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, 6452009);