#7766554

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

6708 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)(5,6,11,13)(7,12,17,18)(8,14,15,9)(10,16,19,20);(1,2)(3,4)(5,7,11,17)(6,12,13,18)(8,16,15,20)(9,10,14,19)(21,22,26,27)(23,28,33,36)(24,29,34,32)(25,30,35,31);(6,13)(7,17)(8,15)(10,19)(22,27)(25,35)(28,36)(29,32);(5,8)(6,14)(7,16)(9,13)(10,18)(11,15)(12,19)(17,20)(21,23)(22,28)(24,30)(25,32)(26,33)(27,36)(29,35)(31,34);(5,9,11,14)(6,15,13,8)(7,10,17,19)(12,20,18,16)(21,24)(22,29)(23,31)(25,28)(26,34)(27,32)(30,33)(35,36);(5,10)(6,16)(7,14)(8,18)(9,17)(11,19)(12,15)(13,20)(21,25)(22,30)(23,32)(24,28)(26,35)(27,31)(29,33)(34,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7673750);