#237918

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

794 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,21)(14,17)(18,24)(19,22)(20,23)(25,26)(27,32)(28,39)(29,44)(30,35)(31,34)(33,41)(36,53)(37,49)(38,47)(40,46)(42,51)(43,52)(45,48)(50,54)(55,56);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,22)(17,23)(21,24)(25,27)(26,32)(28,40)(29,37)(30,45)(31,38)(33,50)(34,47)(35,48)(36,42)(39,46)(41,54)(43,55)(44,49)(51,53)(52,56);(11,18)(12,13)(15,22)(16,21)(19,20)(23,24)(25,28,31,43)(26,33,34,51)(27,36,38,54)(29,42,30,41)(32,46,47,56)(35,52,44,39)(37,55,45,40)(48,50,49,53);(9,12,14,13)(10,16,17,21)(11,19,20,18)(15,22,23,24)(26,34)(32,47)(33,51)(35,44)(39,52)(46,56)(48,49)(50,53)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,f^c=fh,g^c=ghi

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, 145114);