#137968

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#32939	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

612 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 96 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)(7,17,14,27)(8,13,22,29)(9,24,26,20)(10,16)(11,15)(12,19)(18,33)(21,34)(23,30,28,35)(25,36)(31,32);(5,7,17,6,12,20,26,15,10,14,27,16,11,24,9,19)(8,18,32,13,28,34,21,30,22,31,33,29,23,36,25,35);(2,4)(5,8)(6,13)(7,18)(9,25)(10,22)(11,23)(12,28)(14,31)(15,30)(16,29)(17,32)(19,35)(20,34)(21,26)(24,36)(27,33)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,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, 45164);