#7683367

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

8254 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590563);