#6629999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628847
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34641	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3156 conjugacy classes of subgroups; only the 762 normal subgroups are shown.

normal subgroups only

This group has 3156 conjugacy classes of subgroups — too many to draw. Only its 762 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) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,c^b=ci,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537195);