#227124

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

732 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134320);