#192881

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,e^b=eh,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322077, #72322157, #72322190
maximal normal subgroup of	#72322077, #72322157, #72322190

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100077);