#431457

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322536
maximal normal subgroup of	#72322536

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338653);