#249455

order	512
nilpotency class	3
isoclinism	isoclinic class of #249427
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1191 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1191 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156651);