#191888

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320023, #72320103, #72320111, …
maximal normal subgroup of	#72320023, #72320103, #72320111, …
maximal quotient of	#72321588

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99084);