#6440079

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2148 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)(15,23)(16,21)(17,19)(18,31)(20,26)(22,30)(24,39)(25,38)(27,41)(28,34)(29,44)(32,33)(35,40)(36,42)(37,43);(1,2)(3,4)(6,9)(10,12)(14,19)(20,33)(21,34)(22,31)(23,38)(35,36)(37,44)(39,41);(2,4)(13,15,17,25)(14,20,19,33)(16,27,28,24)(18,32,30,26)(21,36,34,35)(22,38,31,23)(29,40,43,42)(37,39,44,41);(2,4)(13,16,17,28)(14,21,19,34)(15,24,25,27)(18,29,30,43)(20,35,33,36)(22,37,31,44)(23,39,38,41)(26,40,32,42);(5,7,8,11)(6,10,9,12)(16,28)(21,34)(24,27)(29,43)(35,36)(37,44)(39,41)(40,42)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,g^c=gi,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, 6347275);