#246307

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
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	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

1351 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)(7,11)(8,10)(9,12)(13,14,18,16)(15,22,24,28)(17,25,27,21)(19,26,20,23);(1,2)(3,4)(6,10)(7,8)(11,12)(13,15,18,24)(14,19,16,20)(17,26,27,23)(21,28,25,22);(2,4)(13,16,18,14)(15,19,24,20)(17,21,27,25)(22,23,28,26);(5,7)(8,9)(10,11)(13,15)(14,20)(16,19)(17,26)(18,24)(21,22)(23,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153503);