#7658228

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

8164 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)(6,9)(7,11)(15,23)(20,30)(24,37)(25,38)(33,41)(34,42)(39,40)(43,44);(1,2)(3,4)(6,9)(10,12)(14,19)(15,23)(21,31)(22,32)(24,37)(25,38)(35,36)(39,40);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,30)(24,33)(25,34)(26,35)(27,31)(28,36)(29,32)(37,41)(38,42)(39,43)(40,44);(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,24)(17,25)(18,23)(19,30)(21,33)(22,34)(26,39)(27,37)(28,40)(29,38)(31,41)(32,42)(35,43)(36,44);(13,16,18,27)(14,21,19,31)(15,24,23,37)(17,28,29,26)(20,33,30,41)(22,36,32,35)(25,40,38,39)(34,44,42,43);(13,17,18,29)(14,22,19,32)(15,25,23,38)(16,26,27,28)(20,34,30,42)(21,35,31,36)(24,39,37,40)(33,43,41,44)

Polycyclic presentation:

2^9|e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565424);