#8077402

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

6578 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)(7,11)(8,12)(9,19)(10,14)(13,17)(15,20)(16,18)(21,22,26,28)(23,24,30,32)(25,27,34,36)(29,31,33,35);(1,2,3,4)(5,7)(6,11)(8,17)(9,16)(10,15)(12,13)(14,20)(18,19)(21,23)(22,24)(25,33)(26,30)(27,35)(28,32)(29,34)(31,36);(7,15)(9,17)(11,20)(13,19)(21,24,26,32)(22,23,28,30)(25,31,34,35)(27,29,36,33);(5,8)(6,12)(7,9)(10,16)(11,19)(13,20)(14,18)(15,17)(21,25)(22,27)(23,29)(24,31)(26,34)(28,36)(30,33)(32,35);(5,9)(6,13)(7,16)(8,15)(10,17)(11,12)(14,19)(18,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);(5,6)(7,11)(8,18)(9,13)(10,14)(12,16)(15,20)(17,19)(21,23)(22,24)(25,33)(26,30)(27,35)(28,32)(29,34)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7984598);