#7651229

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

7458 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,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, 7558425);