#377136

order	512
nilpotency class	3
isoclinism	isoclinic class of #25798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14189
prime radical	#23412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319466, #72319965, #72342961, …
maximal normal subgroup of	#72319466, #72319965, #72342961, …
maximal quotient of	#72343045

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284332);