#412171

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60643	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1574 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=h,e^2=i,f^2=i,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320029, #72320063, #72320109, …
maximal normal subgroup of	#72320029, #72320063, #72320109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319367);