#7714428

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

6408 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621624);