#343341

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

871 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=hi,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,f^c=fi,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, 250537);