#7846812

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

6794 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)(5,6,11,9)(7,12,10,14)(8,13,17,18)(15,19,16,20)(21,22,27,29)(23,28,33,36)(24,30,31,25)(26,32,35,34);(1,2)(3,4)(5,7)(6,12)(8,15)(9,14)(10,11)(13,19)(16,17)(18,20)(21,23,27,33)(22,28,29,36)(24,34,31,32)(25,35,30,26);(7,10)(12,14)(15,16)(19,20)(22,29)(23,33)(24,31)(26,35);(5,8)(6,13)(7,15)(9,18)(10,16)(11,17)(12,19)(14,20)(21,24)(22,30)(23,34)(25,29)(26,28)(27,31)(32,33)(35,36);(5,9,11,6)(7,14,10,12)(8,13,17,18)(15,19,16,20)(21,25,27,30)(22,31,29,24)(23,35,33,26)(28,32,36,34);(5,10,11,7)(6,14,9,12)(8,16,17,15)(13,20,18,19)(21,26,27,35)(22,32,29,34)(23,30,33,25)(24,28,31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=i,f^2=i,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, 7754008);