#7726975

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

7045 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)(5,6,10,7)(8,13,15,11)(9,12,17,14)(16,19,20,18)(21,22)(23,28)(24,25)(26,31)(27,29)(30,36)(32,33)(34,35);(1,2)(3,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,23,27,26)(22,28,29,31)(24,32,30,34)(25,33,36,35);(6,7)(8,15)(12,14)(16,20)(22,29)(24,30)(28,31)(32,34);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,24,27,30)(22,25,29,36)(23,32,26,34)(28,33,31,35);(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,25)(22,30)(23,33)(24,29)(26,35)(27,36)(28,34)(31,32);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,26,27,23)(22,31,29,28)(24,34,30,32)(25,35,36,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

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, 7634171);