#7645904

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

7104 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)(5,6)(7,12)(8,10)(9,11)(14,18)(15,21)(20,25)(22,27);(1,2)(3,4)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28);(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(6,11)(10,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(5,6)(7,10)(8,12)(9,11);(5,8,9,7)(6,12,11,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553100);