#346450

order	512
nilpotency class	3
isoclinism	isoclinic class of #346446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2441 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 2441 conjugacy classes of subgroups — too many to draw. Only its 196 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271366, #72271368, #72271370
maximal normal subgroup of	#72271366, #72271368, #72271370

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253646);