#7927128

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	#38948	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,e^2=gi,f^2=i,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, 7834324);