#246494

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(6,10)(7,8)(11,12)(13,14,19,24)(15,25,16,28)(17,23,18,22)(20,27,21,26);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15,19,16)(14,20,24,21)(17,27,18,26)(22,28,23,25);(2,4)(5,7,9,8)(6,10,11,12)(13,15,19,16)(14,21,24,20)(17,27,18,26)(22,25,23,28);(5,6)(7,10)(8,12)(9,11)(13,16,19,15)(14,21,24,20)(17,26,18,27)(22,25,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153690);