#494534

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224792, #72226794, #72226795, …
maximal normal subgroup of	#72224792, #72226794, #72226795, …
maximal quotient of	C₄×#13025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401730);