#7657746

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×#3518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6460 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)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35);(1,2,3,4)(5,6,10,8)(7,11,15,13)(9,12,17,16)(14,18,20,19)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35);(5,8,10,6)(7,13,15,11)(9,12,17,16)(14,18,20,19);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)

Polycyclic presentation:

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

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, 7564942);