#193516

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂²×C₂²⋊C₄
prime radical	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1420 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)(6,10,11,12)(13,14,17,21,18,22,16,25)(15,20,23,28,24,27,26,19);(5,6,7,10,9,11,8,12)(13,15)(14,19)(16,23)(17,26)(18,24)(20,25)(21,27)(22,28);(2,4)(13,15)(14,20)(16,26)(17,23)(18,24)(19,25)(21,28)(22,27);(5,7,9,8)(6,10,11,12)(13,16,18,17)(14,21,22,25)(15,23,24,26)(19,27,28,20)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100712);