#394598

order	512
nilpotency class	3
isoclinism	isoclinic class of #394598
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14718
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60367	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2164 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)(3,9)(4,15)(5,19)(6,11)(7,13)(8,18)(10,21)(12,24)(14,26)(16,29)(17,25)(20,23)(22,27)(28,31)(30,32);(1,3)(2,7)(4,16)(5,20)(6,14)(8,22)(9,19)(10,26)(11,23)(12,18)(13,21)(15,28)(17,27)(24,30)(25,29)(31,32);(1,4,5,17,6,18,21,32)(2,8,10,25,11,15,19,30)(3,12,13,27,14,16,20,31)(7,22,9,24,23,28,26,29);(1,3)(2,9)(4,12)(5,13)(6,14)(7,19)(8,24)(10,23)(11,26)(15,29)(16,18)(17,27)(20,21)(22,30)(25,28)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215938, #72219687, #72219689, …
maximal normal subgroup of	#72215938, #72219687, #72219689, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301794);