#405852

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14919
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

949 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338517, #72338521, #72338528
maximal normal subgroup of	#72338517, #72338521, #72338528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313048);