#6667547

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38916	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2905 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)(7,11)(8,10)(9,12)(13,14,17,19)(15,23,25,20)(16,26,27,21)(18,22,24,28);(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(18,24)(20,26)(21,23)(25,27);(2,4)(13,15,17,25)(14,20,19,23)(16,18,27,24)(21,22,26,28);(6,10)(9,11)(14,19)(20,23)(21,26)(22,28);(7,12)(9,11)(13,16,17,27)(14,21,19,26)(15,24,25,18)(20,28,23,22)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281696, #72281760
maximal normal subgroup of	#72281696, #72281760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574743);