#246228

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37679	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1400 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:

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

Polycyclic presentation:

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

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, 153424);