#7673548

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

6837 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=g,f^2=g,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, 7580744);