#8037026

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

7285 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)(7,8)(9,17)(10,11)(12,15)(13,18)(14,19)(16,20)(22,26)(24,32)(27,29)(30,33);(1,2,3,4)(6,11)(8,12)(13,17)(16,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,34)(33,35);(5,7,10,15)(6,12,11,8)(9,14,18,20)(13,19,17,16)(21,22,25,26)(23,27,31,29)(24,28,32,34)(30,35,33,36);(5,7)(6,8)(9,14)(10,15)(11,12)(13,16)(17,19)(18,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36);(5,8)(6,7)(9,16)(10,12)(11,15)(13,14)(17,20)(18,19)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,32)(34,35);(5,9)(6,13)(7,14)(8,16)(10,18)(11,17)(12,19)(15,20)(21,24)(22,28)(23,30)(25,32)(26,34)(27,35)(29,36)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7944222);