#244920

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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	#10304	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1182 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1182 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)(7,12)(8,16)(9,11)(10,14)(13,15)(17,19)(18,20)(22,27)(23,31)(24,26)(28,32)(33,34)(35,36);(6,11)(7,9)(12,14)(13,16)(15,18)(19,20)(21,22)(23,32)(24,29)(25,27)(26,30)(28,33)(31,35)(34,36);(2,4)(5,7,10,9)(6,12,14,11)(8,15,17,18)(13,19,20,16)(21,23,25,34)(22,28,27,35)(24,33,26,31)(29,36,30,32);(5,8,10,17)(6,13,14,20)(7,15,9,18)(11,16,12,19)(21,24,25,26)(22,29,27,30)(23,33,34,31)(28,36,35,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152116);