#7711827

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

6968 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22)(23,28)(24,29)(25,27)(26,33)(30,34)(31,36)(32,35);(1,2)(3,4)(21,23,27,24)(22,28,25,29)(26,31,34,32)(30,35,33,36);(6,10)(11,13)(12,17)(18,19)(22,25)(23,24)(30,33)(31,32);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,27,23)(22,29,25,28)(26,32,34,31)(30,36,33,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25,27,22)(23,29,24,28)(26,33,34,30)(31,36,32,35);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26,27,34)(22,30,25,33)(23,31,24,32)(28,35,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619023);