#195456

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
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	#60949	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1083 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:

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

Polycyclic presentation:

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

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, 102652);