#7649901

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

6574 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)(5,6,10,13)(7,11,14,18)(8,9,12,17)(15,16,19,20)(21,22,27,32)(23,28,35,29)(24,34,33,30)(25,31,36,26);(1,2,3,4)(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20)(21,23,27,35)(22,28,32,29)(24,25,33,36)(26,34,31,30);(7,14)(11,18)(15,19)(16,20)(21,23,27,35)(22,29,32,28)(24,25,33,36)(26,30,31,34);(5,8)(6,9)(7,15)(10,12)(11,16)(13,17)(14,19)(18,20)(21,24)(22,30)(23,25)(26,28)(27,33)(29,31)(32,34)(35,36);(5,9)(6,12)(7,16)(8,13)(10,17)(11,19)(14,20)(15,18)(21,25)(22,31)(23,33)(24,35)(26,32)(27,36)(28,30)(29,34);(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20)(21,26,27,31)(22,25,32,36)(23,34,35,30)(24,28,33,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7557097);