#7824391

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

6440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,f^2=h,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, 7731587);