#394804

order	512
nilpotency class	3
isoclinism	isoclinic class of #394803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14719
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1590 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)(3,7)(4,15)(5,10)(6,14)(8,11)(9,18)(12,17)(13,21)(16,23)(19,27)(20,25)(22,24)(26,31)(28,30)(29,32);(1,3,7,2,8,14,6,11)(4,9,22,24,18,15,17,12)(5,13,21,10,20,27,19,25)(16,23,31,32,30,28,29,26);(1,4,6,17,8,18,7,22)(2,9,3,12,11,15,14,24)(5,16,19,29,20,30,21,31)(10,23,13,26,25,28,27,32);(1,5)(2,10)(3,13)(4,16)(6,19)(7,21)(8,20)(9,23)(11,25)(12,26)(14,27)(15,28)(17,29)(18,30)(22,31)(24,32)

Polycyclic presentation:

2^9|b^2=ei,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=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	#72215931, #72219687, #72219691
maximal normal subgroup of	#72215931, #72219687, #72219691

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302000);