#7652437

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6520 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)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(22,23)(27,29)(28,30)(34,35);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,22,26,23)(24,27,32,29)(25,28,33,30)(31,34,36,35);(7,13)(11,14)(15,19)(18,20)(21,22,26,23)(24,27,32,29)(25,30,33,28)(31,35,36,34);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559633);