#394407

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
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	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2136 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302322, #72302839, #72302866
maximal normal subgroup of	#72302322, #72302839, #72302866

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301603);