#6444563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2234 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351759);