#343073

order	512
nilpotency class	3
isoclinism	isoclinic class of #343051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

663 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 663 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,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, 250269);