#7527396

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2205 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,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,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434592);