#101482

order	512
nilpotency class	3
isoclinism	isoclinic class of #101351
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

355 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=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, 8678);