#7895572

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#42478	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7802768);