#193778

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100974);