#8036988

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

7442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,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, 7944184);