#427819

order	512
nilpotency class	3
isoclinism	isoclinic class of #427819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15375
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1412 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)(4,8)(5,9)(6,10)(11,21)(12,24)(13,20)(14,28)(15,23)(16,19)(17,25)(18,27)(22,30)(26,32)(29,31);(1,3)(2,7)(4,11)(5,16)(6,13)(8,21)(9,24)(10,20)(12,17)(14,29)(15,27)(18,23)(19,25)(22,31)(26,30)(28,32);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,22)(10,23)(12,26)(13,27)(16,29)(17,30)(19,31)(20,21)(24,32)(25,28);(1,5)(2,9)(3,12,13,16)(4,14)(6,17)(7,19,20,24)(8,22)(10,25)(11,26,27,29)(15,30)(18,31,21,32)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237762, #72237763, #72237767, …
maximal normal subgroup of	#72237762, #72237763, #72237767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335015);