#392829

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14696
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1464 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282115, #72306313, #72306459, …
maximal normal subgroup of	#72282115, #72306313, #72306459, …
maximal quotient of	#72343044

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300025);