#376400

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

975 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)(8,10)(9,12)(13,14)(15,25)(16,30)(17,20)(18,23)(19,24)(21,28)(22,31)(26,34)(27,35)(29,38)(32,36)(33,37)(39,40)(41,43)(42,44);(5,7,8,12)(6,9,10,11)(14,20)(15,26)(17,18)(21,34)(22,30)(23,24)(27,40)(28,29)(31,33)(35,43)(36,37)(41,42);(13,15,19,29)(14,21,24,34)(16,27,32,41)(17,26,18,28)(20,25,23,38)(22,35,37,43)(30,39,36,44)(31,42,33,40);(1,3,4,2)(6,10)(9,11)(13,16)(14,22)(15,27)(17,33)(18,31)(19,32)(20,30)(21,35)(23,36)(24,37)(25,39)(26,40)(28,42)(29,41)(34,43)(38,44)

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=i,d^2=h,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, 283596);