#7712593

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

7974 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)(6,9)(7,10)(14,16)(18,19);(1,2)(3,4)(5,6,11,9)(7,8,10,12)(13,14,17,16)(15,18,20,19);(5,7,11,10)(6,12,9,8)(13,14,17,16)(15,19,20,18);(5,8)(6,10)(7,9)(11,12)(13,15,17,20)(14,18,16,19);(5,9,11,6)(7,12,10,8)(13,16,17,14)(15,19,20,18);(5,10,11,7)(6,12,9,8)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262490, #72262499, #72262524
maximal normal subgroup of	#72262490, #72262499, #72262524

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619789);