#440846

order	512
nilpotency class	3
isoclinism	isoclinic class of #440846
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338550, #72338552, #72338561
maximal normal subgroup of	#72338550, #72338552, #72338561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348042);