#7935917

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

6681 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)(7,15)(8,17)(9,14)(10,12)(11,16)(13,18)(19,20)(24,32)(29,34)(31,33)(35,36);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,20)(21,22,27,26)(23,30,25,28)(24,29,32,34)(31,36,33,35);(6,12)(7,16)(13,17)(14,20)(21,23,27,25)(22,28,26,30)(24,31,32,33)(29,35,34,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,20)(21,24,27,32)(22,29,26,34)(23,31,25,33)(28,35,30,36);(5,8,10,18)(6,13,12,17)(7,14,16,20)(9,15,19,11)(21,25,27,23)(22,28,26,30)(24,33,32,31)(29,35,34,36);(5,9)(6,14)(7,13)(8,11)(10,19)(12,20)(15,18)(16,17)(21,26,27,22)(23,30,25,28)(24,34,32,29)(31,36,33,35)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7843113);