#6380526

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250423, #72256890
maximal normal subgroup of	#72250423, #72256890
maximal quotient of	#72264214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287722);