#192037

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1020 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285976, #72306420, #72306430, …
maximal normal subgroup of	#72285976, #72306420, #72306430, …
maximal quotient of	#72287453, #72287454

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99233);