#264682

order	512
nilpotency class	3
isoclinism	isoclinic class of #264659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171878);