#7733800

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

8832 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)(13,14)(15,18)(16,21)(17,20)(19,24)(22,25)(23,27)(26,28);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(14,15)(17,25)(19,21)(23,28);(6,9)(10,11)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(5,6)(7,10)(8,9)(11,12);(5,6)(7,10)(8,9)(11,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^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	#72304580, #72304593, #72304639
maximal normal subgroup of	#72304580, #72304593, #72304639

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640996);