#6500145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3540	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3204 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:

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

Polycyclic presentation:

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

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, 6407341);