#557014

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1902 conjugacy classes of subgroups; only the 858 normal subgroups are shown.

normal subgroups only

This group has 1902 conjugacy classes of subgroups — too many to draw. Only its 858 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) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fh,d^2=fhi,e^2=hi,b^a=bf,c^a=cg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 464210);