#101086

order	512
nilpotency class	3
isoclinism	isoclinic class of #100939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

922 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 922 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gh,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8282);