#6545088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#3307	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2010 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)(13,14,19,22,18,24,17,23)(15,20,26,35,28,36,27,34)(16,29,30,21,32,38,31,37)(25,39,41,33,42,43,40,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,35)(23,34)(24,36)(29,39)(30,41)(31,40)(32,42)(37,44)(38,43);(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,29)(23,38)(24,37)(26,40)(27,41)(28,42)(34,43)(35,39)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,24,23)(15,26,28,27)(16,30,32,31)(20,34,36,35)(21,29,37,38)(25,40,42,41)(33,43,44,39);(7,11)(10,12)(13,17,18,19)(14,22,24,23)(15,27,28,26)(16,30,32,31)(20,35,36,34)(21,29,37,38)(25,41,42,40)(33,39,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452284);