#347633

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

923 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 923 conjugacy classes of subgroups — too many to draw. Only its 164 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

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, 254829);