#138077

order	512
nilpotency class	4
isoclinism	isoclinic class of #125422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 80 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)(7,17,21,24)(8,13)(9,26,28,23)(10,15)(11,16)(12,20)(14,18,19,33)(22,34,29,35)(25,31)(27,32)(30,36);(5,7,18,31,27,22,9,20,11,21,33,36,10,29,28,13)(6,12,26,34,15,25,19,17,16,8,23,35,32,30,14,24);(2,4)(5,8,10,25,11,12,27,30)(6,13,15,31,16,20,32,36)(7,19,22,23,21,14,29,26)(9,24,18,35,28,17,33,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45273);