#152998

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator, socle, prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

246 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322704, #72322863, #72322876
maximal normal subgroup of	#72322704, #72322863, #72322876
maximal quotient of	#72322720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60194);