#7655555

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

6875 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=i,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, 7562751);