#374819

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

957 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)(8,9)(11,12)(13,14)(15,25)(16,20)(17,23)(18,24)(19,28)(21,30)(22,34)(26,39)(27,36)(29,37)(31,40)(32,41)(33,38)(35,42)(43,44);(13,15,18,29)(14,19,24,38)(16,27,32,42)(17,33,34,28)(20,39,41,44)(21,36,40,35)(22,37,23,25)(26,30,43,31);(1,3,4,2)(6,9)(10,12)(13,16)(14,20)(15,26)(17,30)(18,32)(19,35)(21,23)(22,40)(24,41)(25,39)(27,33)(28,42)(29,43)(31,34)(36,38)(37,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,27)(17,31)(18,32)(19,36)(20,23)(22,41)(24,40)(25,39)(26,33)(28,43)(29,42)(30,34)(35,38)(37,44)

Polycyclic presentation:

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

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, 282015);