#345923

order	512
nilpotency class	3
isoclinism	isoclinic class of #345888
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

488 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253119);