#140989

order	512
nilpotency class	4
isoclinism	isoclinic class of #125393
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,f^c=fi,f^d=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, 48185);