#6382152

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	#34935	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2390 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)(13,14,18,22,17,21,16,20)(15,19,25,28,24,27,23,26);(5,6,9,10)(7,11,8,12)(13,14,16,20,17,21,18,22)(15,19,23,26,24,27,25,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,24)(14,19,21,27)(16,23,18,25)(20,26,22,28);(5,8,9,7)(6,11,10,12)(13,16,17,18)(14,20,21,22)(15,23,24,25)(19,26,27,28);(15,24)(19,27)(23,25)(26,28)

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,d^b=dfi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307586, #72307595, #72307613
maximal normal subgroup of	#72307586, #72307595, #72307613

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289348);