#472074

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

972 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=i,d^2=ei,e^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,d^b=di,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379270);