#7657007

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×#3524	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6496 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)(7,16)(8,14)(9,10)(11,13)(12,15)(17,20)(18,19)(21,22,27,26)(23,31,34,28)(24,29,33,35)(25,30,32,36);(1,2,3,4)(5,7,12,11)(6,13,15,16)(8,19,10,17)(9,18,14,20);(5,8,12,10)(6,14,15,9)(7,17,11,19)(13,18,16,20)(21,23)(22,28)(24,32)(25,33)(26,31)(27,34)(29,30)(35,36);(5,9)(6,8)(7,18)(10,15)(11,20)(12,14)(13,19)(16,17)(21,24,27,33)(22,29,26,35)(23,32,34,25)(28,30,31,36);(5,10,12,8)(6,9,15,14)(7,17,11,19)(13,18,16,20)(21,25,27,32)(22,30,26,36)(23,33,34,24)(28,29,31,35);(5,11,12,7)(6,13,15,16)(8,19,10,17)(9,20,14,18)(21,26,27,22)(23,31,34,28)(24,35,33,29)(25,30,32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=h,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, 7564203);