#6382247

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2400 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)(6,10)(11,12)(13,14,18,23,17,22,19,26)(15,20,24,27,16,21,25,28);(5,6,9,10)(7,11,8,12)(13,14,19,26,17,22,18,23)(15,21,25,27,16,20,24,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,16)(14,20,22,21)(18,24,19,25)(23,27,26,28);(5,8,9,7)(6,11,10,12)(13,16,17,15)(14,21,22,20)(18,25,19,24)(23,28,26,27);(15,16)(20,21)(24,25)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216212
maximal normal subgroup of	#72216212
maximal quotient of	#72215932

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289443);