#413185

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35001	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2816 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2816 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=h,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2022, #72310626, #72310652
maximal normal subgroup of	D₄×#2022, #72310626, #72310652

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320381);