#6447652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354848);