#181627

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1357 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:

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

Polycyclic presentation:

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

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, 88823);