#6473021

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6380217);