#7643206

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

8834 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)(6,9)(7,11)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,27)(25,26);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,19,16)(14,20,21,22)(17,24,23,26)(18,25,28,27);(5,6)(7,10)(8,9)(11,12)(14,21)(17,23)(20,22)(24,26);(5,7)(6,10)(8,11)(9,12)(13,16,19,15)(14,22,21,20)(17,26,23,24)(18,27,28,25);(13,17,19,23)(14,18,21,28)(15,24,16,26)(20,25,22,27);(13,18)(14,23)(15,25)(16,27)(17,21)(19,28)(20,26)(22,24)

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^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.

Relationships

maximal subgroup of	#72315000
maximal normal subgroup of	#72315000

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550402);