#7645826

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6452 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)(6,10)(7,13)(12,18)(14,15)(22,24)(23,29)(28,32)(31,35);(1,2)(3,4)(21,22,26,24)(23,30,29,27)(25,28,33,32)(31,36,35,34);(5,6)(7,11)(8,12)(9,10)(13,17)(14,19)(15,20)(16,18)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(5,7,9,13)(6,11,10,17)(8,15,16,14)(12,20,18,19)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,8,9,16)(6,12,10,18)(7,14,13,15)(11,19,17,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36);(5,6)(7,11)(8,12)(9,10)(13,17)(14,19)(15,20)(16,18)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=h,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, 7553022);