#6629458

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2276 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,e^b=ei,e^c=ei,h^d=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536654);