#326622

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2598
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

431 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fi

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, 233818);