#7653796

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

8308 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)(14,15)(19,21)(20,22)(27,28);(1,2,3,4)(5,6)(7,10)(8,11)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28);(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27);(5,7)(6,10)(8,12)(9,11)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27);(5,6)(7,9)(8,11)(10,12)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304538, #72304595, #72304623
maximal normal subgroup of	#72304538, #72304595, #72304623

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7560992);