#412989

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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

980 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)(15,27,31,36)(16,23,34,39)(17,22,33,40)(18,25,19,24)(21,30,37,29)(28,35,43,44)(32,41,42,38);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,32)(17,28)(18,30)(19,29)(20,31)(22,38)(23,35)(24,36)(25,27)(26,37)(33,43)(34,42)(39,44)(40,41);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,33,20,34,19,17)(14,22,24,39,26,40,25,23)(15,28,29,42,31,43,30,32)(21,35,27,41,37,44,36,38);(13,17,19,34,20,33,18,16)(14,23,25,40,26,39,24,22)(15,28,29,42,31,43,30,32)(21,35,27,41,37,44,36,38)

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320185);