#7720249

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

6924 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)(7,12)(8,10)(9,17)(11,13)(14,18)(15,19)(16,20)(21,22,27,24)(23,29,25,28)(26,30,34,32)(31,36,33,35);(1,2)(3,4)(5,7,11,8)(6,12,13,10)(9,16,18,15)(14,19,17,20)(21,23,27,25)(22,28,24,29)(26,31,34,33)(30,35,32,36);(6,13)(7,8)(9,18)(19,20)(22,24)(28,29)(30,32)(35,36);(5,8,11,7)(6,10,13,12)(9,16,18,15)(14,19,17,20)(21,24,27,22)(23,29,25,28)(26,32,34,30)(31,36,33,35);(5,9,11,18)(6,14,13,17)(7,15,8,16)(10,19,12,20)(21,25,27,23)(22,29,24,28)(26,33,34,31)(30,36,32,35);(5,10)(6,7)(8,13)(9,19)(11,12)(14,15)(16,17)(18,20)(21,26,27,34)(22,30,24,32)(23,31,25,33)(28,35,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 7627445);