#7718453

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

6648 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)(8,16)(11,18)(14,20)(17,19)(22,23)(25,33)(27,29)(31,36);(1,2)(3,4)(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,26,23)(24,27,32,29)(25,28,33,30)(31,34,36,35);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,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, 7625649);