#7720242

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

6946 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)(7,12)(8,16)(9,14)(10,11)(13,18)(15,17)(19,20)(21,22)(23,30)(24,28)(25,27)(26,33)(29,34)(31,35)(32,36);(1,2,3,4)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(6,11)(7,14)(13,16)(15,20)(22,25)(23,24)(29,33)(31,32);(5,7,10,14)(6,12,11,9)(8,15,18,20)(13,19,16,17)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,25,27,22)(23,28,24,30)(26,33,34,29)(31,35,32,36);(5,9,10,12)(6,14,11,7)(8,17,18,19)(13,20,16,15)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7627438);