#8073134

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

6828 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)(7,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,20)(21,22,26,28)(23,29,31,25)(24,27,33,36)(30,32,35,34);(1,2)(3,4)(5,7,10,11)(6,8,13,15)(9,14,18,19)(12,16,17,20)(21,23)(22,25)(24,30)(26,31)(27,34)(28,29)(32,36)(33,35);(7,11)(8,15)(14,19)(16,20)(21,23,26,31)(22,25,28,29)(24,30,33,35)(27,34,36,32);(5,6)(7,8)(9,17)(10,13)(11,15)(12,18)(14,20)(16,19)(21,24)(22,27)(23,30)(25,34)(26,33)(28,36)(29,32)(31,35);(5,8)(6,11)(7,13)(9,16)(10,15)(12,19)(14,17)(18,20)(21,23)(22,25)(24,30)(26,31)(27,34)(28,29)(32,36)(33,35);(5,9)(6,12)(7,14)(8,16)(10,18)(11,19)(13,17)(15,20)(21,25)(22,23)(24,32)(26,29)(27,35)(28,31)(30,36)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^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, 7980330);