#6371192

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,27)(17,20)(18,22)(19,33)(21,30)(23,32)(25,39)(26,35)(28,41)(29,42)(31,38)(34,43)(36,44)(37,40);(6,9)(10,12)(13,15,18,17)(14,20,22,24)(16,25,29,28)(19,26,32,31)(21,34,36,37)(23,35,33,38)(27,41,42,39)(30,40,44,43);(13,16,19,30,18,29,32,44)(14,21,23,27,22,36,33,42)(15,25,26,40,17,28,31,43)(20,34,35,41,24,37,38,39);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,20,22,24)(16,28,29,25)(19,31,32,26)(21,34,36,37)(23,35,33,38)(27,41,42,39)(30,43,44,40);(15,17)(16,29)(20,24)(21,36)(26,31)(27,42)(30,44)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278388);