#7527550

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45104	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2028 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,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^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, 7434746);