#195794

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60693	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

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, 102990);