#334849

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	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

822 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

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, 242045);