#193571

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	C₂×#3106	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1378 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100767);