#481750

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35144	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

963 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318176, #72318640, #72318754
maximal normal subgroup of	#72318176, #72318640, #72318754

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388946);