#359105

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249848, #72249991, #72252967, …
maximal normal subgroup of	#72249848, #72249991, #72252967, …
maximal quotient of	#72265468

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266301);