#254273

order	512
nilpotency class	3
isoclinism	isoclinic class of #254227
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	C₂³	C₂×#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=i,d^2=i,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,d^c=di,e^c=eh,f^c=fi,g^c=ghi,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, 161469);