#6467552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28354	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,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, 6374748);