#7744216

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7651412);