#7658627

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

7216 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)(8,17)(9,12)(10,13)(14,20)(15,18)(16,19)(24,32)(27,33)(30,36)(34,35);(1,2)(3,4)(5,7,10,16)(6,11,13,19)(8,14,9,15)(12,18,17,20)(21,22)(23,28)(24,33)(25,29)(26,31)(27,32)(30,34)(35,36);(5,8,10,9)(6,12,13,17)(7,14,16,15)(11,18,19,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,36)(33,35);(5,7)(6,11)(8,14)(9,15)(10,16)(12,18)(13,19)(17,20)(21,24)(22,27)(23,30)(25,32)(26,34)(28,35)(29,33)(31,36);(5,9,10,8)(6,12,13,17)(7,15,16,14)(11,18,19,20)(21,23)(22,28)(24,30)(25,31)(26,29)(27,35)(32,36)(33,34);(5,6)(7,11)(8,12)(9,17)(10,13)(14,18)(15,20)(16,19)(21,22)(23,26)(24,27)(25,29)(28,31)(30,34)(32,33)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565823);