#244114

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151310);