#244643

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1031 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1031 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,11)(4,12)(5,8)(6,9)(7,10)(13,15)(14,16)(17,18,22,27)(19,28,31,42)(20,24,33,41)(21,25,35,26)(23,34,37,30)(29,44,46,43)(32,39,48,40)(36,45,38,47);(1,3)(2,4)(5,13)(6,12)(7,14)(8,11)(9,16)(10,15)(17,19)(18,23)(20,29)(21,34)(22,31)(24,38)(25,28)(26,42)(27,37)(30,35)(32,47)(33,46)(36,41)(39,44)(40,43)(45,48);(2,8)(3,12)(5,6)(9,10)(11,16)(13,14)(17,20,22,33)(18,24,27,41)(19,29,31,46)(21,32,35,48)(23,36,37,38)(25,39,26,40)(28,43,42,44)(30,45,34,47);(1,4)(2,3)(5,16)(6,11)(7,15)(8,12)(9,13)(10,14)(18,25)(21,35)(23,28)(24,39)(26,27)(30,34)(32,48)(36,43)(37,42)(38,44)(40,41)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151839);