#6457365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2344 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6364561);