#452648

order	512
nilpotency class	3
isoclinism	isoclinic class of #427241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16057
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

816 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)(3,13,17,8)(4,18,21,28)(5,11)(6,12)(7,10)(9,20,27,22)(14,30)(15,25,16,24)(19,23)(26,31)(29,32);(1,3,6,16,7,17,5,15)(2,8,12,25,10,13,11,24)(4,19,20,31,21,32,22,14)(9,26,18,29,27,30,28,23);(1,4,5,20,7,21,6,22)(2,9,12,28,10,27,11,18)(3,14,15,19,17,31,16,32)(8,23,25,30,13,29,24,26);(8,13)(23,29)(24,25)(26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251699, #72251700, #72251705, …
maximal normal subgroup of	#72251699, #72251700, #72251705, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359844);