#511855

order	512
nilpotency class	4
isoclinism	isoclinic class of #511853
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32899
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,e^c=eh,g^d=gi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240644, #72249650, #72249656, …
maximal normal subgroup of	#72240644, #72249650, #72249656, …
maximal quotient of	#72255673, #72255918, #72262119, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419051);