#189034

order	512
nilpotency class	3
isoclinism	isoclinic class of #188956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh,f^c=fh,g^c=ghi

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, 96230);