#403372

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14873
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2328 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)(3,14)(4,19)(5,13)(6,11)(7,12)(8,16)(9,17)(10,20)(15,29)(18,24)(21,27)(22,26)(23,31)(25,32)(28,30);(1,3)(2,8,12,24)(4,15)(5,16)(6,18)(7,17)(9,11,14,13)(10,23,26,29)(19,28,27,25)(20,30)(21,31)(22,32);(1,3)(2,9,12,14)(4,15)(5,16)(6,18)(7,17)(8,13,24,11)(10,25,26,28)(19,23,27,29)(20,30)(21,31)(22,32);(1,4)(2,10)(3,15)(5,20)(6,22)(7,21)(8,23)(9,25)(11,19)(12,26)(13,27)(14,28)(16,30)(17,31)(18,32)(24,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279719, #72285055, #72285150, …
maximal normal subgroup of	#72279719, #72285055, #72285150, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310568);