#192520

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

817 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99716);