#7739370

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

7782 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(14,19)(17,25)(20,21)(23,24);(6,7)(10,11)(13,14)(15,21)(16,20)(17,22)(18,19)(23,28)(24,27)(25,26);(13,15,18,16)(14,20,19,21)(17,23,25,24)(22,27,26,28);(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,21,19,20)(17,24,25,23)(22,28,26,27);(5,8,9,12)(6,10,7,11)(13,17,18,25)(14,22,19,26)(15,23,16,24)(20,27,21,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7646566);