#320562

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 190 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224718, #72224725, #72224726
maximal normal subgroup of	#72224718, #72224725, #72224726

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227758);