#135719

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#334
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=e,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#135719, #72210514, #72210518
maximal normal subgroup of	C₂×#135719, #72210514, #72210518
maximal quotient of	#72210465, C₂×#135719

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42915);