#6448964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23391	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2654 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283332, #72283747, #72283829
maximal normal subgroup of	#72283332, #72283747, #72283829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356160);