#277598

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

756 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)(13,14)(15,21)(16,25)(17,19)(18,23)(20,22)(24,27)(26,28);(5,6)(7,9)(8,11)(10,12)(13,15,16,23,17,24,22,28)(14,18,20,21,19,26,25,27);(2,4)(6,9)(7,12)(10,11)(14,19)(18,26)(20,25)(21,27);(13,15,22,28,17,24,16,23)(14,18,25,27,19,26,20,21)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184794);