#413934

order	512
nilpotency class	3
isoclinism	isoclinic class of #394043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2613
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15159
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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)(3,14)(4,10)(5,11)(6,22)(7,12)(8,13)(9,16)(15,30)(17,19)(18,32)(20,28)(21,27)(23,26)(24,25)(29,31);(1,3)(2,9)(4,13)(5,15)(6,10)(7,16)(8,17)(11,24)(12,19)(14,22)(18,27)(20,32)(21,31)(23,25)(26,29)(28,30);(1,4)(2,10)(3,12)(5,18)(6,19)(7,14)(8,9)(11,25)(13,16)(15,28)(17,22)(20,29)(21,24)(23,30)(26,31)(27,32);(1,5,7,20,8,21,6,23)(2,11,12,28,13,27,22,26)(3,15,16,32,17,31,10,25)(4,18,14,29,9,24,19,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338482, #72338485, #72338500, …
maximal normal subgroup of	#72338482, #72338485, #72338500, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321130);