#151902

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1398, #72319593, #72319601, …
maximal normal subgroup of	D₄×#1398, #72319593, #72319601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59098);