#6457169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2311 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)(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(2,4)(5,7)(6,9)(8,11)(10,12);(7,11)(9,12)(13,15,16,23,17,22,25,26)(14,18,19,21,20,27,24,28);(7,11)(9,12);(15,22)(18,27)(21,28)(23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283745, #72283931, #72316272, …
maximal normal subgroup of	#72283745, #72283931, #72316272, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364365);