#389271

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

828 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efh,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

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, 296467);