#245098

order	512
nilpotency class	3
isoclinism	isoclinic class of #245084
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#34988	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1689 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)(3,8)(4,14)(5,17)(6,10)(7,11)(9,12)(13,22)(15,28)(16,21)(18,25)(19,30)(20,26)(23,29)(24,27)(31,32);(1,3,7,5)(2,8,11,17)(4,12,16,18)(6,13,20,19)(9,23,25,28)(10,22,26,30)(14,24,21,32)(15,27,29,31);(1,4,7,16)(2,9,11,25)(3,12,5,18)(6,15,20,29)(8,21,17,14)(10,24,26,32)(13,27,19,31)(22,23,30,28);(3,13)(5,19)(8,22)(12,27)(14,28)(17,30)(18,31)(21,23)

Polycyclic presentation:

2^9|b^2=i,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286785
maximal normal subgroup of	#72286785

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152294);