#7792786

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

7766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(2,4)(13,14,17,18)(15,16,19,20);(1,2,3,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11)(14,18)(15,19);(7,12)(10,11)(13,15)(14,16)(17,19)(18,20);(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,20)(18,19);(5,6)(7,10)(8,9)(11,12)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7699982);