#135226

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1467
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20,16,21,32,38,15,22,31,37,13,19,28,36)(11,23,24,39,26,17,30,33,25,34,29,40,12,18,27,35);(2,5)(6,8)(9,11,13,12,15,25,16,26)(10,17,21,34,22,18,19,23)(14,29,28,30,31,24,32,27)(20,35,38,39,37,33,36,40);(1,3,4,7)(2,6,5,8)(9,12,13,25,15,26,16,11)(10,18,19,34,22,17,21,23)(14,30,28,24,31,27,32,29)(20,35,36,40,37,33,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240695, #72240697, #72240700
maximal normal subgroup of	#72240695, #72240697, #72240700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42422);