#7813031

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	#46354	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4603 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7720227);