#7711501

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
prime radical	#39383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6566 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)(7,11)(8,16)(9,14)(10,12)(13,18)(15,20)(17,19)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,34)(35,36);(1,2,3,4)(5,7,10,9)(6,11,12,14)(8,15,18,17)(13,19,16,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(6,12)(11,14)(13,16)(19,20)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(5,6)(7,11)(8,13)(9,14)(10,12)(15,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,8,10,18)(6,13,12,16)(7,15,9,17)(11,19,14,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,9,10,7)(6,14,12,11)(8,17,18,15)(13,20,16,19)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,e^2=hi,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, 7618697);