#191872

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320669, #72320684, #72320729
maximal normal subgroup of	#72320669, #72320684, #72320729
maximal quotient of	#72321184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99068);