#245515

order	512
nilpotency class	3
isoclinism	isoclinic class of #245447
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	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

947 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 947 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152711);