#7630509

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

8906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

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, 7537705);