#7626248

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3490	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8142 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533444);