#511797

order	512
nilpotency class	4
isoclinism	isoclinic class of #511796
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#23910	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2362 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 2362 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418993);