#6463609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292956, #72293019, #72293024, …
maximal normal subgroup of	#72292956, #72293019, #72293024, …
maximal quotient of	C₂×#6463609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370805);