#7658959

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

7000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566155);