#244522

order	512
nilpotency class	3
isoclinism	isoclinic class of #244499
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

1031 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1031 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

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, 151718);