#473870

order	512
nilpotency class	3
isoclinism	isoclinic class of #453995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2812
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16885
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338916, #72338919, #72338926
maximal normal subgroup of	#72338916, #72338919, #72338926

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381066);