#8070105

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

6785 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7977301);