#318061

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
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

710 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)(13,14,18,22)(15,20,23,28)(16,25,24,19)(17,27,26,21);(5,6,7,10,8,11,12,9)(13,15,17,24,18,23,26,16)(14,19,21,20,22,25,27,28);(2,4)(6,9)(7,12)(10,11)(15,23)(16,24)(19,25)(20,28);(13,16,26,23,18,24,17,15)(14,20,27,19,22,28,21,25)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,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^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225257);