#7712369

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

6646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=h,f^2=i,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, 7619565);