#138658

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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×#2213	C₂
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

389 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,4)(5,6,10,14,25,15,17,33,11,13,28,27,9,8,23,24)(7,16,20,32,21,12,29,30,19,18,34,35,22,31,26,36);(5,7,17,34,9,21,10,26,11,19,23,20,25,22,28,29)(6,12,27,35,15,16,24,30,13,31,14,32,8,18,33,36);(2,4)(5,8)(6,9)(7,18)(10,27)(11,15)(12,21)(13,25)(14,28)(16,19)(17,33)(20,30)(22,31)(23,24)(26,35)(29,32)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216405, #72216411, #72216412
maximal normal subgroup of	#72216405, #72216411, #72216412
maximal quotient of	#72226914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45854);