#246766

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	#45976	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1328 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,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, 153962);