#488611

order	512
nilpotency class	4
isoclinism	isoclinic class of #488523
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3186	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

758 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh

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, 395807);