#275604

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ei,c^2=ei,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 182800);