#7504174

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#31057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1949 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)(7,8)(9,11)(10,12)(13,14,19,24,20,25,18,23)(15,17,21,16,22,26,28,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,20,22)(14,17,25,26)(16,23,27,24)(18,28,19,21);(2,4)(13,16,20,27)(14,21,25,28)(15,23,22,24)(17,18,26,19);(6,11)(7,12)(13,17)(14,22)(15,25)(16,18)(19,27)(20,26)(21,24)(23,28);(5,8,9,10)(6,7,11,12)(13,18,20,19)(14,23,25,24)(15,21,22,28)(16,26,27,17)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

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, 7411370);