#454457

order	512
nilpotency class	3
isoclinism	isoclinic class of #454057
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, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1594 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1594 conjugacy classes of subgroups — too many to draw. Only its 214 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361653);