#7503810

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	#23422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2282 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7411006);