#115904

order	512
nilpotency class	3
isoclinism	isoclinic class of #96924
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

(2,4)(9,10,12,14,13,16,11,15)(17,18,21,24,20,22,28,30)(19,25,27,32,26,23,29,31);(1,2,3,4)(5,6)(7,8)(18,22)(23,25)(24,30)(31,32);(6,8)(10,14)(11,12)(15,16)(17,19,20,26)(18,23,22,25)(21,29,28,27)(24,32,30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,c^b=cf,e^c=ei,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23100);