#192573

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#385
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99769);