#321712

order	512
nilpotency class	3
isoclinism	isoclinic class of #278235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=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, 228908);