#194304

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#3100	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101500);