#7643360

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

7450 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)(7,11)(10,12)(13,14)(15,16)(17,25)(18,19)(20,23)(21,26)(22,27)(24,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(14,19)(16,20)(21,25)(24,27);(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,25)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,15,19,23)(17,24,26,27)(21,22,25,28);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,21,19,25)(15,22,23,28)(16,24,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550556);