#479828

order	512
nilpotency class	3
isoclinism	isoclinic class of #478248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1029 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387024);