#417859

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314443, #72314448, #72314468
maximal normal subgroup of	#72314443, #72314448, #72314468
maximal quotient of	#72285528, #72314512

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325055);