#412992

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,12)(8,10)(9,11)(13,14,20,26)(15,27,30,40)(16,23,33,38)(17,22,32,39)(18,25,19,24)(21,34,36,29)(28,41,42,35)(31,37,43,44);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,20,30)(14,21,26,36)(16,31,33,43)(17,28,32,42)(18,34,19,29)(22,37,39,44)(23,35,38,41)(24,40,25,27);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,17,20,33,19,32)(14,22,24,23,26,39,25,38)(15,28,29,31,30,42,34,43)(21,35,27,37,36,41,40,44);(13,17,18,33,20,32,19,16)(14,23,24,39,26,38,25,22)(15,28,34,43,30,42,29,31)(21,35,40,44,36,41,27,37)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320188);