#137160

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

714 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=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, 44356);