#138041

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

764 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fh,g^b=gi,h^b=hi,d^c=di,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, 45237);