#7806366

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
prime radical	#37620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6558 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)(21,22,26,28)(23,29,30,24)(25,33,34,27)(31,32,36,35);(1,2,3,4)(6,11)(7,8)(9,17)(19,20)(23,30)(24,29)(25,34)(27,33);(5,6)(7,12)(8,13)(9,18)(10,11)(14,17)(15,20)(16,19)(21,23,26,30)(22,24,28,29)(25,31,34,36)(27,32,33,35);(5,7,10,8)(6,12,11,13)(9,15,17,16)(14,19,18,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8,10,7)(6,13,11,12)(9,16,17,15)(14,20,18,19)(21,25,26,34)(22,27,28,33)(23,31,30,36)(24,32,29,35);(5,9)(6,14)(7,15)(8,16)(10,17)(11,18)(12,19)(13,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=hi,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, 7713562);