#7680652

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

8682 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)(3,4)(7,11)(10,12)(13,14,17,19)(15,16,20,18);(5,6,8,9)(7,10,11,12)(13,15)(14,18)(16,19)(17,20);(2,4)(6,9)(10,12)(13,16,17,18)(14,15,19,20);(1,2)(3,4)(13,15)(14,16)(17,20)(18,19);(5,6)(7,10)(8,9)(11,12);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7587848);