#6605833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38948	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,40)(20,34,37,43)(21,26,22,27)(24,32,41,31)(25,38,44,36)(30,39,35,42)(45,47,48,46);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32,23,33,21,31)(18,24,26,40,28,41,27,29)(20,35,36,47,37,30,38,46)(25,42,34,45,44,39,43,48);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,21,38,23,37,22,36)(18,25,27,43,28,44,26,34)(19,30,31,47,33,35,32,46)(24,39,29,45,41,42,40,48);(9,11)(10,13)(12,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,40,41,29)(25,43,44,34)(30,46,35,47)(39,45,42,48);(11,15)(13,16)(20,37)(25,44)(30,35)(34,43)(36,38)(39,42)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi

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, 6513029);