#345904

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

502 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,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, 253100);