#495715

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2192	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402, #72216443, #72216448, …
maximal normal subgroup of	#72216402, #72216443, #72216448, …
maximal quotient of	#72225208, C₂×#495715

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402911);