#122040

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1446
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

693 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=ei,g^c=gi

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, 29236);