#7630812

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

8142 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)(6,10)(11,12)(13,14)(15,24)(16,22)(17,19)(18,21)(20,23)(25,28)(26,27);(5,6)(7,11)(8,12)(9,10)(13,15,20,18)(14,21,23,24)(16,27,17,25)(19,26,22,28);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,16,20,17)(14,22,23,19)(15,25,18,27)(21,26,24,28);(5,7,9,8)(6,11,10,12)(13,17,20,16)(14,19,23,22)(15,25,18,27)(21,26,24,28);(5,8,9,7)(6,12,10,11)(13,18,20,15)(14,21,23,24)(16,27,17,25)(19,28,22,26);(13,19)(14,16)(15,26)(17,23)(18,28)(20,22)(21,27)(24,25)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538008);