#7604426

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=h,d^2=g,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,e^d=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, 7511622);