#7647858

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

6918 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)(8,16)(12,19)(14,20)(17,18)(23,26)(24,32)(28,30)(29,34);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,15)(14,18)(16,19)(17,20)(21,22,27,25)(23,28,26,30)(24,29,32,34)(31,35,33,36);(5,7)(6,10)(8,14)(9,15)(11,13)(12,17)(16,20)(18,19)(21,23,27,26)(22,28,25,30)(24,31,32,33)(29,35,34,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,24)(22,29)(23,31)(25,34)(26,33)(27,32)(28,35)(30,36);(5,6)(7,10)(8,12)(9,13)(11,15)(14,17)(16,19)(18,20)(21,25,27,22)(23,30,26,28)(24,29,32,34)(31,35,33,36);(5,8,9,16)(6,12,13,19)(7,14,15,20)(10,17,11,18)(21,26,27,23)(22,30,25,28)(24,33,32,31)(29,36,34,35)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=h,f^2=hi,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, 7555054);