#195036

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1167 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,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, 102232);