#7763295

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	C₂×#3545	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6606 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:

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

Polycyclic presentation:

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

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, 7670491);