#495373

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321767, #72321852, #72321861
maximal normal subgroup of	#72321767, #72321852, #72321861

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402569);