#204669

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1196 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)(6,9,10,11)(13,14,17,21,18,22,16,20)(15,23,26,27,24,28,25,19);(5,6,7,11,8,10,12,9)(13,15)(14,19,22,27)(16,25)(17,26)(18,24)(20,28,21,23);(1,2,3,4)(15,24)(19,27)(23,28)(25,26);(13,16,18,17)(14,20,22,21)(15,25,24,26)(19,28,27,23)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111865);