#345996

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=efi,c^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253192);