#7643207

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

8850 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)(5,6,8,10)(7,11,12,9)(13,14,19,17)(15,21,16,20)(18,25,26,22)(23,27,24,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,16)(14,20,17,21)(18,23,26,24)(22,27,25,28);(5,7)(6,9)(8,12)(10,11)(14,17)(20,21)(22,25)(27,28);(5,6)(7,9)(8,10)(11,12)(13,16,19,15)(14,21,17,20)(18,24,26,23)(22,28,25,27);(13,17,19,14)(15,21,16,20)(18,25,26,22)(23,28,24,27);(13,18)(14,22)(15,23)(16,24)(17,25)(19,26)(20,27)(21,28)

Polycyclic presentation:

2^9|a^2=g,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, 7550403);