#460624

order	512
nilpotency class	3
isoclinism	isoclinic class of #460624
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16368
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1120 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)(3,7,13,20)(4,8)(5,16,17,9)(6,10)(11,21,26,18)(12,27)(14,22,30,24)(15,23)(19,29)(25,32)(28,31);(1,3)(2,7,10,20)(4,11)(5,12)(6,13)(8,21,23,18)(9,19,16,27)(14,25)(15,26)(17,29)(22,32,24,31)(28,30);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,24)(10,23)(12,28)(13,26)(16,22)(17,30)(19,31)(20,21)(25,29)(27,32);(1,5)(2,9)(3,12)(4,14)(6,17)(7,19)(8,22)(10,16)(11,25)(13,29)(15,30)(18,31)(20,27)(21,32)(23,24)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237740, #72237763, #72237768, …
maximal normal subgroup of	#72237740, #72237763, #72237768, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367820);