#7660792

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×#3516	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6527 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)(21,22,27,30)(23,24,25,28)(26,29,34,36)(31,32,33,35);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,25)(22,24,30,28)(26,31,34,33)(29,32,36,35);(6,10)(11,17)(12,18)(19,20)(21,23,27,25)(22,28,30,24)(26,31,34,33)(29,35,36,32);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,24,27,28)(22,23,30,25)(26,32,34,35)(29,31,36,33);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,23)(22,28,30,24)(26,31,34,33)(29,32,36,35);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,26,27,34)(22,29,30,36)(23,31,25,33)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7567988);