#6456552

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	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282435, #72282518, #72282860
maximal normal subgroup of	#72282435, #72282518, #72282860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363748);