#388013

order	512
nilpotency class	3
isoclinism	isoclinic class of #32700
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2760
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14508
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

812 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284899, #72286147, #72286154, …
maximal normal subgroup of	#72284899, #72286147, #72286154, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295209);