#433299

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15476
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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)(3,12,14,7)(4,16)(5,9)(6,10)(8,18)(11,21)(13,22,17,25)(15,24,20,23)(19,27)(26,30)(28,31,29,32);(1,3)(2,7)(4,17,18,13)(5,20)(6,14)(8,25,16,22)(9,24)(10,12)(11,23)(15,21)(19,28,30,29)(26,31,27,32);(1,4)(2,8,10,16)(3,13)(5,19)(6,18)(7,22,12,25)(9,26,11,27)(14,17)(15,28)(20,29)(21,30)(23,31,24,32);(2,9,10,11)(7,23,12,24)(8,26,16,27)(22,31,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285237, #72286050, #72286134, …
maximal normal subgroup of	#72285237, #72286050, #72286134, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340495);