#496420

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ei,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312126, #72312151, #72312165, …
maximal normal subgroup of	#72312126, #72312151, #72312165, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403616);