#495845

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33580	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237289, #72239538, #72239543, …
maximal normal subgroup of	#72237289, #72239538, #72239543, …
maximal quotient of	#72318532, #72318661, #72319826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403041);