#340007

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247203);