#394408

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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2316 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)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,18)(19,27)(20,25)(21,28);(2,4)(13,15)(14,19)(16,22)(17,26)(18,27)(20,28)(21,25)(23,24);(17,24)(20,25)(21,28)(23,26)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,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	#72234964, #72234968, #72234972
maximal normal subgroup of	#72234964, #72234968, #72234972
maximal quotient of	#72236944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301604);