#410850

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	#15091
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 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)(4,20)(5,10)(6,12)(7,13)(8,17)(9,22)(11,25)(15,31)(16,21)(18,28)(19,27)(23,24)(26,32)(29,30);(1,3,6,18,7,19,25,17)(2,8,12,14,13,28,11,27)(4,21,10,15,23,30,22,32)(5,16,24,31,9,29,20,26);(1,4,6,10,7,23,25,22)(2,9,12,20,13,5,11,24)(3,15,17,21,19,32,18,30)(8,26,27,29,28,31,14,16);(1,5)(2,10)(3,16)(4,11)(6,24)(7,9)(8,15)(12,23)(13,22)(14,30)(17,26)(18,31)(19,29)(20,25)(21,27)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338511, #72338520, #72338528
maximal normal subgroup of	#72338511, #72338520, #72338528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318046);