#7646753

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	#37440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6772 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7553949);