#7199103

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#24257	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1965 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106299);