#140228

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

281 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^c=egh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216405, #72216411, #72216414
maximal normal subgroup of	#72216405, #72216411, #72216414
maximal quotient of	#72226926

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47424);