#181244

order	512
nilpotency class	3
isoclinism	isoclinic class of #181127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1370 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=hi,g^2=hi,b^a=be,e^a=eh,g^a=ghi,f^b=fi,d^c=dh

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, 88440);