#7654755

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

6910 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)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20);(1,2)(3,4)(22,26)(27,33)(28,34)(35,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,22,25,26)(23,27,30,33)(24,28,32,34)(29,35,31,36);(5,8,9,17)(6,11,13,16)(7,14,15,20)(10,18,12,19);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561951);