#410823

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3104 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 3104 conjugacy classes of subgroups — too many to draw. Only its 250 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264408, #72264512, #72264521
maximal normal subgroup of	#72264408, #72264512, #72264521

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318019);