#249405

order	512
nilpotency class	3
isoclinism	isoclinic class of #249391
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

1205 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1205 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(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);(4,13)(6,15)(7,12)(10,14)(17,19,22,29)(18,23,26,36)(20,27,31,41)(21,28,33,43)(24,34,38,39)(25,35,32,42)(30,40,44,48)(37,46,45,47);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(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);(17,21)(18,25)(19,28)(20,30)(22,33)(23,35)(24,37)(26,32)(27,40)(29,43)(31,44)(34,46)(36,42)(38,45)(39,47)(41,48)

Polycyclic presentation:

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

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, 156601);