#6672299

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3012	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1641 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579495);