#7987867

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

6848 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)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22,26,31)(23,27,33,28)(24,34,35,29)(25,30,36,32);(1,2,3,4)(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,23)(22,27)(24,32)(25,34)(26,33)(28,31)(29,36)(30,35);(6,8)(11,13)(12,16)(18,19)(21,23,26,33)(22,28,31,27)(24,30,35,32)(25,29,36,34);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,23)(22,28)(24,32)(25,29)(26,33)(27,31)(30,35)(34,36);(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,24)(22,29)(23,32)(25,28)(26,35)(27,36)(30,33)(31,34);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,25)(22,30)(23,29)(24,27)(26,36)(28,35)(31,32)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895063);