#263220

order	512
nilpotency class	3
isoclinism	isoclinic class of #263219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265388
maximal normal subgroup of	#72265388

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170416);