#6372744

order	512
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#120
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×C₄×#120
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72236792, #72236967, #72240986, …
maximal subgroup of	#72236792, #72236845, #72236873, …
maximal normal subgroup of	#72236792, #72236873, #72236884, …
maximal quotient of	#72251096, C₂×#6372744, #72262427, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279940);