#263396

order	512
nilpotency class	3
isoclinism	isoclinic class of #263395
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

923 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)(4,10)(5,9)(6,8)(7,13)(12,16)(14,15)(17,18,24,19)(20,28,30,32)(21,22,26,23)(25,31,27,29);(1,3,5,13)(2,7,9,11)(4,12,6,14)(8,16,10,15)(17,19,22,26,24,18,23,21)(20,25,29,28,30,27,31,32);(2,8)(4,6)(7,15)(9,10)(11,16)(12,14)(17,20)(18,25)(19,27)(21,28)(22,29)(23,31)(24,30)(26,32);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,16,11,15)(17,21,22,19,24,26,23,18)(20,28,29,27,30,32,31,25)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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=ghi,c^b=cg,d^b=dh,f^b=fhi,d^c=dhi,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305433, #72305445, #72305464
maximal normal subgroup of	#72305433, #72305445, #72305464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170592);