#7542042

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 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)(7,10)(8,12)(9,11)(13,14,18,20)(15,16,24,26)(17,28,27,22)(19,25,23,21);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15,17,21,18,24,27,25)(14,19,22,16,20,23,28,26);(5,8,9,10)(6,7,11,12)(13,14)(15,23)(16,25)(17,22)(18,20)(19,24)(21,26)(27,28);(2,4)(6,11)(7,12)(14,20)(16,26)(19,23)(22,28);(13,16,27,19,18,26,17,23)(14,21,28,15,20,25,22,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=gi,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,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, 7449238);