#6817304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2015 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724500);