#7630511

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

8150 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)(6,11)(7,12)(17,25)(21,26)(22,27)(23,28);(5,6,10,11)(7,8,12,9)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(14,19)(15,20)(21,26)(22,27);(5,8,10,9)(6,12,11,7)(13,15,18,20)(14,16,19,24)(17,22,25,27)(21,23,26,28);(5,9,10,8)(6,7,11,12)(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,26)(25,28);(13,17)(14,21)(15,22)(16,23)(18,25)(19,26)(20,27)(24,28)

Polycyclic presentation:

2^9|b^2=g,d^2=gi,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, 7537707);