#394288

order	512
nilpotency class	3
isoclinism	isoclinic class of #394287
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14716
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2614 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)(3,6)(4,13)(5,10)(7,11)(8,12)(9,18)(14,20)(15,17)(16,29)(19,24)(21,25)(22,26)(23,30)(27,31)(28,32);(1,3,7,12,8,11,6,2)(4,15,18,24,19,9,17,13)(5,14,21,26,22,25,20,10)(16,28,32,29,31,23,30,27);(1,4)(2,9)(3,13)(5,16)(6,17)(7,18)(8,19)(10,23)(11,24)(12,15)(14,27)(20,30)(21,32)(22,31)(25,29)(26,28);(1,5)(2,10)(3,14)(4,16)(6,20)(7,21)(8,22)(9,23)(11,25)(12,26)(13,27)(15,28)(17,30)(18,32)(19,31)(24,29)

Polycyclic presentation:

2^9|b^2=ei,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,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	#72213423, #72219485, #72219488, …
maximal normal subgroup of	#72213423, #72219485, #72219488, …
maximal quotient of	#72255683, C₂×#394288

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301484);