#7533678

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#59037	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2044 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,23,26,27)(16,21,17,20)(19,25,28,24);(1,2)(3,4)(7,12)(9,11)(13,15,16,25,18,26,17,24)(14,19,20,27,22,28,21,23);(5,7,8,12)(6,9,10,11)(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,23,28,27);(2,4)(6,10)(9,11)(13,16,18,17)(14,20,22,21)(15,25,26,24)(19,27,28,23);(13,15,16,25,18,26,17,24)(14,19,20,27,22,28,21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440874);