#245386

order	512
nilpotency class	3
isoclinism	isoclinic class of #245375
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

1065 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1065 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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,g^b=gh,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152582);