#421916

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15299
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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)(3,5,12,18)(4,8,16,22)(6,10)(7,9)(11,15)(13,19,26,31)(14,21)(17,23,30,28)(20,24)(25,29)(27,32);(1,3)(2,5,9,18)(4,14,16,27)(6,13)(7,12)(8,21)(10,19,24,31)(11,17,25,30)(15,28)(20,26)(22,32)(23,29);(1,4)(2,8,9,22)(3,11,12,25)(5,15)(6,17)(7,16)(10,23,24,28)(13,14,26,27)(18,29)(19,21)(20,30)(31,32);(1,5)(2,3)(4,15)(6,19)(7,18)(8,11)(9,12)(10,13)(14,23)(16,29)(17,21)(20,31)(22,25)(24,26)(27,28)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286176, #72286218, #72286242
maximal normal subgroup of	#72286176, #72286218, #72286242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329112);