#7901933

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

6638 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)(7,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,20)(21,22)(23,31)(24,33)(25,30)(26,28)(27,32)(29,34)(35,36);(1,2)(3,4)(21,23)(22,27)(24,30)(25,29)(26,32)(28,31)(33,35)(34,36);(6,11)(7,12)(13,18)(14,19)(22,28)(23,32)(29,33)(30,36);(5,7,10,12)(6,8,11,16)(9,14,17,19)(13,15,18,20)(21,23)(22,27)(24,30)(25,29)(26,32)(28,31)(33,35)(34,36);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(21,24,26,34)(22,29,28,33)(23,30,32,36)(25,31,35,27);(5,9,10,17)(6,13,11,18)(7,14,12,19)(8,15,16,20)(21,25)(22,30)(23,29)(24,27)(26,35)(28,36)(31,34)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7809129);