#7719227

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

6840 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)(9,17)(12,16)(14,20)(18,19)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,34)(35,36);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,17)(19,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(6,8)(11,13)(12,16)(18,19)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626423);