#246790

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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	#25725	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1371 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153986);