#363561

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1010 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)(13,14,19,24,20,25,18,33)(15,23,17,32,28,40,27,21)(16,22,30,38,31,39,34,37)(26,36,41,35,42,44,29,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,27,20,28,19,17)(14,21,24,23,25,32,33,40)(16,26,30,41,31,42,34,29)(22,35,37,36,39,43,38,44);(2,4)(6,9)(10,12)(13,16,18,30,20,31,19,34)(14,22,24,37,25,39,33,38)(15,26,27,41,28,42,17,29)(21,35,23,36,32,43,40,44);(5,7,8,11)(6,10,9,12)(13,17,19,28,20,27,18,15)(14,23,33,21,25,40,24,32)(16,29,34,42,31,41,30,26)(22,36,38,35,39,44,37,43)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=egi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270757);