#443525

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups; only the 262 normal subgroups are shown.

normal subgroups only

This group has 1638 conjugacy classes of subgroups — too many to draw. Only its 262 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 — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=gi,d^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=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, 350721);