#427668

order	512
nilpotency class	3
isoclinism	isoclinic class of #427668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15371
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

987 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)(4,14,17,8)(5,18)(6,10)(9,19)(11,24,15,20)(12,26)(13,22)(16,23,30,28)(21,27)(25,31,29,32);(1,3)(2,7)(4,15,17,11)(5,12)(6,13)(8,20,14,24)(9,21)(10,22)(16,25,30,29)(18,26)(19,27)(23,32,28,31);(1,4)(2,8)(3,11)(5,16)(6,17)(7,20)(9,23)(10,14)(12,25)(13,15)(18,28)(19,30)(21,31)(22,24)(26,32)(27,29);(1,5)(2,9,10,18)(3,12)(4,16)(6,19)(7,21,22,26)(8,23,14,28)(11,25)(13,27)(15,29)(17,30)(20,31,24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237785, #72237786, #72237793, …
maximal normal subgroup of	#72237785, #72237786, #72237793, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334864);