#366130

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273326);