#244532

order	512
nilpotency class	3
isoclinism	isoclinic class of #244523
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1148 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 151728);