#7557204

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#24792	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1840 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=g,e^2=f,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^b=ei,d^c=di,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, 7464400);