#6467581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	#25727	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2281 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6374777);