#183025

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1020 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,14)(15,16)(17,18,22,27,23,28,21,26)(19,20,29,24,30,32,25,31);(1,3,6,14,7,15,5,13)(2,8,9,16,4,12,10,11)(17,19,22,29,23,30,21,25)(18,24,27,32,28,31,26,20);(1,4,7,2)(3,12,15,8)(5,9,6,10)(11,13,16,14)(17,20,21,31,23,32,22,24)(18,25,26,30,28,29,27,19);(1,5,7,6)(2,9,4,10)(3,13,15,14)(8,16,12,11)(17,21,23,22)(18,26,28,27)(19,25,30,29)(20,31,32,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,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, 90221);