#495552

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

868 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)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,30)(18,33)(19,23)(20,28)(22,31)(25,37)(26,38)(27,39)(29,44)(32,34)(35,41)(36,42)(40,43);(2,4)(13,15,25,41,18,29,17,28,19,27,31,43,16,26,32,42)(14,20,34,44,21,35,22,24,23,36,30,38,33,40,37,39);(6,9)(10,12)(13,16,19,18)(14,21,23,33)(15,26,27,29)(17,25,32,31)(20,35,36,40)(22,30,37,34)(24,38,39,44)(28,41,42,43);(5,7,8,11)(6,10,9,12)(15,27)(20,36)(24,39)(26,29)(28,42)(35,40)(38,44)(41,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=f,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321161, #72321167, #72321221, …
maximal normal subgroup of	#72321161, #72321167, #72321221, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402748);