#183024

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,e^b=ei,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90220);