#174685

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

639 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

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, 81881);