#368241

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14016
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237692, #72240445, #72240459, …
maximal normal subgroup of	#72237692, #72240445, #72240459, …
maximal quotient of	#72237708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275437);