#105545

order	512
nilpotency class	3
isoclinism	isoclinic class of #94972
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1226 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 246 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)(6,10)(7,9)(11,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,24,26)(25,28);(5,6,9,11,8,12,7,10)(13,15)(14,18)(16,24)(17,23)(19,27)(20,21)(22,25)(26,28);(2,4)(5,7,8,9)(6,11,12,10)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12741);