#7689753

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

8131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596949);