#7730952

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

6546 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)(6,8)(11,13)(12,16)(18,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(1,2)(3,4)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7638148);