#7649155

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

7484 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)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,19)(17,22)(18,20)(23,28)(24,26)(25,27);(1,2)(3,4)(5,6,8,10)(7,9,12,11)(13,15,18,16)(14,19,20,21)(17,23,25,24)(22,26,27,28);(5,6)(7,9)(8,10)(11,12)(14,20)(19,21)(22,27)(26,28);(5,7,8,12)(6,9,10,11)(13,16,18,15)(14,21,20,19)(17,23,25,24)(22,26,27,28);(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28);(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,28)(25,27)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^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, 7556351);