#7733526

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

8498 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)(13,14)(15,22)(16,17)(18,20)(19,21)(23,28)(24,25)(26,27);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,21)(15,22,16,17)(18,26,27,20)(23,25,24,28);(6,9)(10,11)(13,15,19,16)(14,17,21,22)(18,23,27,24)(20,25,26,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,17,21,22)(18,24,27,23)(20,25,26,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,15)(16,21)(18,25)(19,22)(20,23)(24,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,18)(14,20)(15,23)(16,24)(17,25)(19,27)(21,26)(22,28)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304546, #72304590, #72304599
maximal normal subgroup of	#72304546, #72304590, #72304599

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640722);