#451309

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15994
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 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)(3,13,16,8)(4,17)(5,12)(6,10)(7,11)(9,19)(14,23,30,29)(15,25,22,24)(18,26,31,32)(20,28)(21,27);(1,3)(2,8)(4,18,19,31)(5,15)(6,22)(7,16)(9,26,17,32)(10,25)(11,13)(12,24)(14,20,30,21)(23,27,29,28);(1,4)(2,9,11,17)(3,14)(5,21)(6,20)(7,19)(8,23,13,29)(10,27,12,28)(15,31)(16,30)(18,22)(24,32,25,26);(2,10,11,12)(8,24,13,25)(9,27,17,28)(23,32,29,26)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318430, #72318864, #72318963, …
maximal normal subgroup of	#72318430, #72318864, #72318963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358505);