#397631

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

944 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304827);