#285181

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18756	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192377);