#7658127

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

7008 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)(5,6,12,15)(7,16,11,10)(8,13,20,17)(9,18,19,14)(21,22)(23,28)(24,32)(25,29)(26,31)(27,33)(30,34)(35,36);(1,2,3,4)(5,7,12,11)(6,10,15,16)(8,18,20,14)(9,17,19,13)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8)(6,13)(7,14)(9,16)(10,19)(11,18)(12,20)(15,17)(21,24,25,33)(22,27,29,32)(23,30,31,36)(26,34,28,35);(5,9,12,19)(6,14,15,18)(7,17,11,13)(8,16,20,10);(5,10,12,16)(6,7,15,11)(8,19,20,9)(13,14,17,18)(21,23)(22,28)(24,30)(25,31)(26,29)(27,35)(32,34)(33,36);(5,11,12,7)(6,10,15,16)(8,18,20,14)(9,13,19,17)(21,22)(23,26)(24,27)(25,29)(28,31)(30,34)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565323);