#481970

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#34609	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

986 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,24)(20,25)(21,29,30,23)(22,26)(27,32,28,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,23)(19,20)(24,32)(25,31)(26,29)(27,28);(2,8)(7,15)(9,10)(11,16)(17,19,22,20)(18,24,26,25)(21,27,30,28)(23,31,29,32);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,20,22,19)(18,25,26,24)(21,28,30,27)(23,32,29,31)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307969, #72308012
maximal normal subgroup of	#72307969, #72308012

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389166);