#7650254

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

7515 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)(13,14)(15,17)(16,21)(18,19)(20,23)(22,25)(24,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(14,19)(17,20)(21,27)(25,28);(7,11)(10,12)(13,15)(14,20)(16,24)(17,19)(18,23)(21,25)(22,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,22)(17,25)(18,26)(19,27)(20,28)(23,24);(5,6)(7,10)(8,9)(11,12)(13,17,18,20)(14,15,19,23)(16,25,26,28)(21,22,27,24)

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=h,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, 7557450);