#127659

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34855);