#7653882

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

9424 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:

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

Polycyclic presentation:

2^9|a^2=i,b^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	#72304560, #72304580, #72304609
maximal normal subgroup of	#72304560, #72304580, #72304609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561078);