#394600

order	512
nilpotency class	3
isoclinism	isoclinic class of #394600
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14718
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60367	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,13)(4,10)(5,8)(6,11)(7,12)(9,17)(14,23)(15,29)(16,24)(18,21)(19,26)(20,25)(22,30)(27,28)(31,32);(2,8)(3,14)(5,6)(9,13)(10,21)(11,12)(15,28)(18,20)(22,27)(23,24)(25,26)(29,32);(1,3,5,16,7,14,6,17)(2,9,11,23,12,24,8,13)(4,15,18,30,19,28,20,31)(10,22,25,29,26,32,21,27);(1,4)(2,10)(3,15)(5,18)(6,20)(7,19)(8,21)(9,22)(11,25)(12,26)(13,27)(14,28)(16,30)(17,31)(23,29)(24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306445, #72306455, #72306465, …
maximal normal subgroup of	#72306445, #72306455, #72306465, …
maximal quotient of	C₂×#394600, #72316817, #72316921

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301796);