#245022

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1078 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,14)(7,15,8,12)(9,18,19,13)(11,17,20,16)(21,22,25,29)(23,27,30,26)(24,31,32,36)(28,34,35,33);(5,7,10,8)(6,11,14,20)(9,17,19,16)(12,18,15,13)(21,23,25,30)(22,26,29,27)(24,28,32,35)(31,33,36,34);(2,4)(5,8,10,7)(6,12,14,15)(9,17,19,16)(11,18,20,13)(21,24,25,32)(22,28,29,35)(23,31,30,36)(26,34,27,33);(6,13)(9,19)(11,12)(14,18)(15,20)(16,17)(22,29)(23,30)(28,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152218);