#7641883

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

9358 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303257, #72342344
maximal normal subgroup of	#72303257, #72342344
maximal quotient of	#72303294, #72303312, #72304757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549079);