#7657442

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6288 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(1,2,3,4)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35);(5,7,9,14)(6,10,12,18)(8,15,16,13)(11,19,20,17);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 7564638);