#7651203

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

8468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7558399);