#93644

order	512
nilpotency class	4
isoclinism	isoclinic class of #93514
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂³×C₄⋊C₄	C₂²
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄⋊C₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fh,e^2=fh,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72266345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 840);