#355902

order	512
nilpotency class	2
isoclinism	isoclinic class of #180872
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,22,24,35)(13,17)(14,18)(16,23,29,36)(19,31,32,39)(20,27)(21,28)(25,30)(26,33,37,40)(34,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,28)(18,27)(22,31)(23,33)(24,32)(25,34)(29,37)(30,38)(35,39)(36,40);(3,8)(5,7)(9,12)(10,16,18,29)(11,19)(13,23)(14,24)(15,26,27,37)(17,22,30,35)(20,32)(21,33)(25,36)(28,31,38,39)(34,40);(2,6)(5,7)(11,20)(15,27)(19,32)(21,34)(26,37)(28,38)(31,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237199, #72257112, #72257117, …
maximal normal subgroup of	#72237199, #72257112, #72257117, …
maximal quotient of	#72237221, #72257592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263098);