#138299

order	512
nilpotency class	4
isoclinism	isoclinic class of #125300
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

441 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,6,11,15,26,16,9,14,12,17,29,27,10,7,18,20)(8,19,23,34,36,13,21,31,24,32,28,30,22,33,25,35);(5,7,10,6,12,16,26,17)(8,19,22,13,24,32,36,33)(9,14,29,15,18,20,11,27)(21,30,28,31,25,34,23,35);(2,4)(5,8,10,22,12,24,26,36)(6,13,16,32,17,33,7,19)(9,25,11,28,18,21,29,23)(14,34,27,31,20,30,15,35)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=h,d^2=gh,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45495);