#249757

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
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	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156953);