#467840

order	512
nilpotency class	3
isoclinism	isoclinic class of #467840
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16592
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339676, #72339678, #72339684
maximal normal subgroup of	#72339676, #72339678, #72339684

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375036);