#188940

order	512
nilpotency class	3
isoclinism	isoclinic class of #188828
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 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)(17,18,23,28,22,27,21,26)(19,25,29,32,20,24,30,31);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16)(17,19,22,20)(18,24,27,25)(21,30,23,29)(26,32,28,31);(17,20,22,19)(18,25,27,24)(21,30,23,29)(26,32,28,31);(3,11)(7,15)(10,16)(12,13)(17,21,22,23)(18,26,27,28)(19,29,20,30)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96136);