#7763427

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×#3508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6959 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)(5,6,11,10)(7,15,16,12)(8,13,17,19)(9,20,18,14)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,36)(33,34);(1,2)(3,4)(5,7,11,16)(6,12,10,15)(8,18,17,9)(13,20,19,14)(21,22,26,30)(23,28,24,27)(25,29,34,33)(31,36,32,35);(6,10)(7,16)(8,17)(14,20)(21,22)(23,27)(24,28)(25,33)(26,30)(29,34)(31,36)(32,35);(5,8,11,17)(6,13,10,19)(7,9,16,18)(12,14,15,20)(21,23,26,24)(22,27,30,28)(25,31,34,32)(29,35,33,36);(5,9)(6,14)(7,17)(8,16)(10,20)(11,18)(12,19)(13,15)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(5,10,11,6)(7,15,16,12)(8,19,17,13)(9,20,18,14)(21,25,26,34)(22,29,30,33)(23,31,24,32)(27,35,28,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=i,e^2=hi,f^2=i,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, 7670623);