#492906

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×Q₈	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2386 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280674, #72280707, #72280746, …
maximal normal subgroup of	#72280674, #72280707, #72280746, …
maximal quotient of	#72281549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400102);