#7739330

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×#3511	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6702 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:

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

Polycyclic presentation:

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

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, 7646526);