#411519

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224916, #72249848, #72252977, …
maximal normal subgroup of	#72224916, #72249848, #72252977, …
maximal quotient of	#72224913

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318715);