#7720389

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,f^2=i,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, 7627585);