#7630861

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

7112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538057);