#134228

order	512
nilpotency class	4
isoclinism	isoclinic class of #11502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1034 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,f^c=fg,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41424);