#6502710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500120
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6409906);