#7806300

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

6851 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)(23,25)(28,29)(32,35)(33,36);(1,2)(3,4)(5,6)(7,13)(8,16)(9,15)(10,11)(12,18)(14,17)(19,20)(21,22,27,31)(23,29,25,28)(24,34,30,26)(32,33,35,36);(6,11)(9,13)(12,16)(17,19)(21,23,27,25)(22,28,31,29)(24,35,30,32)(26,36,34,33);(5,7)(6,9)(8,14)(10,15)(11,13)(12,17)(16,19)(18,20)(21,24,27,30)(22,26,31,34)(23,32,25,35)(28,33,29,36);(5,8,10,18)(6,12,11,16)(7,14,15,20)(9,17,13,19)(21,25,27,23)(22,29,31,28)(24,35,30,32)(26,36,34,33);(5,9)(6,7)(8,17)(10,13)(11,15)(12,14)(16,20)(18,19)(21,26)(22,30)(23,33)(24,31)(25,36)(27,34)(28,35)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713496);