#480661

order	512
nilpotency class	4
isoclinism	isoclinic class of #480656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,f^b=fg,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285303, #72285386, #72307990
maximal normal subgroup of	#72285303, #72285386, #72307990

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387857);