#7883112

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

6638 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)(5,6,10,7)(8,13,16,11)(9,12,18,14)(15,20,17,19)(21,22,26,25)(23,27,29,31)(24,28,32,33)(30,34,35,36);(1,2)(3,4)(5,6,10,7)(8,11,16,13)(9,12,18,14)(15,19,17,20)(23,29)(27,31)(30,35)(34,36);(6,7)(11,13)(12,14)(19,20)(21,23,26,29)(22,27,25,31)(24,30,32,35)(28,34,33,36);(5,7,10,6)(8,13,16,11)(9,14,18,12)(15,20,17,19);(5,8,10,16)(6,11,7,13)(9,17,18,15)(12,20,14,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,9,10,18)(6,12,7,14)(8,15,16,17)(11,19,13,20)(21,25,26,22)(23,31,29,27)(24,28,32,33)(30,34,35,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7790308);