#7712573

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

6814 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)(8,15)(12,17)(13,19)(18,20)(21,22)(23,25)(24,30)(26,34)(27,28)(29,31)(32,35)(33,36);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(22,28)(23,31)(30,33)(32,34);(6,10)(11,16)(12,17)(18,20)(21,23)(22,29)(24,34)(25,28)(26,33)(27,31)(30,35)(32,36);(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24)(22,30)(23,32)(25,35)(26,29)(27,36)(28,33)(31,34);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,25,27,29)(22,31,28,23)(24,35,36,26)(30,34,33,32);(5,8,9,15)(6,12,10,17)(7,13,14,19)(11,18,16,20)(21,26,27,35)(22,32,28,34)(23,33,31,30)(24,29,36,25)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619769);