#405157

order	512
nilpotency class	3
isoclinism	isoclinic class of #394058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261331, #72284889, #72284933, …
maximal normal subgroup of	#72261331, #72284889, #72284933, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312353);