#7902863

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

7937 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)(5,6)(7,11)(8,9)(10,12)(14,18)(15,21)(20,26)(22,23);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,25)(22,27)(23,28)(24,26);(6,9)(10,11)(13,14,17,18)(15,19,21,25)(16,20,24,26)(22,27,23,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,25)(22,27)(23,28)(24,26);(5,7)(6,10)(8,12)(9,11)(13,15,17,21)(14,19,18,25)(16,23,24,22)(20,28,26,27);(13,16,17,24)(14,20,18,26)(15,22,21,23)(19,27,25,28)

Polycyclic presentation:

2^9|c^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72266767, #72266997, #72266998

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7810059);