#213868

order	512
nilpotency class	3
isoclinism	isoclinic class of #213852
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35355	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1253 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)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18)(19,27,30,23)(20,25,21,24)(22,26)(28,32)(29,31);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27);(6,9)(14,15)(17,20)(18,24)(19,28)(21,22)(23,31)(25,26)(27,32)(29,30);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,21,22,20)(18,25,26,24)(19,29,30,28)(23,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240632, #72240633, #72240635, …
maximal normal subgroup of	#72240632, #72240633, #72240635, …
maximal quotient of	#72251574, #72251584

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121064);