#7714754

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

7832 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)(7,11)(8,10)(9,12)(14,19)(16,27)(17,30)(20,24)(25,35)(26,36)(34,41)(42,44);(1,2)(3,4)(14,19)(15,23)(21,28)(22,31)(25,35)(26,36)(34,41)(39,43);(5,6,8,10)(7,9,12,11)(13,14)(15,24)(16,28)(17,22)(18,19)(20,23)(21,27)(25,32)(26,38)(29,41)(30,31)(33,36)(34,40)(35,37)(39,42)(43,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,25)(17,26)(18,23)(19,24)(21,32)(22,33)(27,35)(28,37)(29,39)(30,36)(31,38)(34,42)(40,43)(41,44);(5,7)(6,9)(8,12)(10,11)(13,16,18,27)(14,21,19,28)(15,25,23,35)(17,29,30,40)(20,32,24,37)(22,34,31,41)(26,39,36,43)(33,42,38,44);(13,17)(14,22)(15,26)(16,29)(18,30)(19,31)(20,33)(21,34)(23,36)(24,38)(25,39)(27,40)(28,41)(32,42)(35,43)(37,44)

Polycyclic presentation:

2^9|c^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,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, 7621950);