#7646657

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6304 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)(5,6,10,9)(7,13,15,11)(8,16,17,12)(14,18,20,19)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35);(1,2)(3,4)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,7)(6,11)(8,14)(9,13)(10,15)(12,18)(16,19)(17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,8,10,17)(6,12,9,16)(7,14,15,20)(11,18,13,19)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35);(5,9,10,6)(7,13,15,11)(8,12,17,16)(14,18,20,19);(5,7)(6,11)(8,14)(9,13)(10,15)(12,18)(16,19)(17,20)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553853);