#213284

order	512
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	C₂³	#532
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

632 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18,22,26,23,27,21,36)(19,28,32,37,29,41,31,24)(20,25,34,39,35,40,33,46)(30,42,45,47,43,48,44,38);(17,19,21,31,23,29,22,32)(18,24,26,28,27,37,36,41)(20,30,33,44,35,43,34,45)(25,38,39,42,40,47,46,48);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(19,29)(24,37)(28,41)(30,43)(31,32)(38,47)(42,48)(44,45);(17,20,22,34,23,35,21,33)(18,25,26,39,27,40,36,46)(19,30,32,45,29,43,31,44)(24,38,28,42,37,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fi,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210557
maximal normal subgroup of	#72210557
maximal quotient of	#72210396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120480);