#6380530

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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

2302 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)(15,23)(16,18)(17,21)(19,22)(20,24)(25,27)(26,28);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,23)(17,24)(18,26)(19,25)(21,27)(22,28);(5,8,9,7)(6,10,11,12)(13,16)(14,18)(15,23)(17,22)(19,21)(20,26)(24,28)(25,27);(13,16)(14,17)(15,20)(18,22)(19,21)(23,26)(24,27)(25,28);(13,17,19,18)(14,21,22,16)(15,24,25,26)(20,27,28,23)

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,d^a=di,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250423, #72256932, #72256959
maximal normal subgroup of	#72250423, #72256932, #72256959
maximal quotient of	#72264214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287726);