#401938

order	512
nilpotency class	3
isoclinism	isoclinic class of #395191
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14825
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285536, #72286225, …
maximal normal subgroup of	#72285525, #72285536, #72286225, …
maximal quotient of	C₂×#401938

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309134);