#105654

order	512
nilpotency class	3
isoclinism	isoclinic class of #105652
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249689, #72249691, #72249700
maximal normal subgroup of	#72249689, #72249691, #72249700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12850);