#7630658

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=gi,e^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537854);