#254194

order	512
nilpotency class	3
isoclinism	isoclinic class of #254179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

756 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,38)(30,46)(31,40)(32,45)(33,41)(34,42)(35,39)(36,43)(37,44)(47,48);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(17,19,22,21)(18,24,26,28)(20,31)(23,30,36,35)(25,40)(27,39,43,46)(29,37)(32,33)(34,47)(38,44)(41,45)(42,48);(1,3,4,13)(2,8,7,15)(5,12,6,14)(9,16,10,11)(17,20)(18,25)(19,29)(21,32)(22,34)(23,37)(24,38)(26,42)(27,44)(28,45)(30,47)(31,35)(33,36)(39,48)(40,46)(41,43);(2,7)(8,15)(9,10)(11,16)(17,21,22,19)(18,24,26,28)(20,32,34,29)(23,35,36,30)(25,38,42,45)(27,39,43,46)(31,33,47,37)(40,44,48,41)

Polycyclic presentation:

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

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, 161390);