#246970

order	512
nilpotency class	3
isoclinism	isoclinic class of #246957
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
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1063 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,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,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,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, 154166);