#7673713

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

8482 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)(6,9)(10,11)(13,14)(15,19)(16,17)(18,20)(21,25)(22,27)(23,24)(26,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,19,20,27)(16,23,21,26)(17,24,25,28);(5,6)(7,11)(8,9)(10,12)(14,20)(15,22)(16,21)(24,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28);(13,16,18,21)(14,17,20,25)(15,23,22,26)(19,24,27,28);(13,17)(14,21)(15,24)(16,20)(18,25)(19,26)(22,28)(23,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580909);