#6440216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#24425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2153 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,19,25,32)(16,21,27,36)(17,20,29,22)(24,31,37,43)(26,30,39,33)(28,34,40,35)(38,41,44,42);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,30)(21,31)(22,33)(23,32)(27,37)(28,38)(29,39)(34,41)(35,42)(36,43)(40,44);(2,4)(14,20)(16,27)(17,29)(19,30)(21,35)(22,23)(24,37)(26,39)(31,42)(32,33)(34,36)(41,43);(2,4)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,35)(23,36)(25,37)(26,38)(29,40)(30,41)(32,43)(33,42)(39,44);(5,7,8,11)(6,10,9,12)(15,25)(16,27)(19,32)(21,36)(26,39)(28,40)(30,33)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347412);