#6462999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2498 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)(7,12)(8,10)(9,11)(13,14,19,23,18,15,17,21)(16,20,26,28,24,22,25,27);(7,8)(10,12)(13,15,18,14)(16,22,24,20)(17,21,19,23)(25,27,26,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,8,9,7)(6,10,11,12);(16,24)(20,22)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308676, #72308809, #72308910
maximal normal subgroup of	#72308676, #72308809, #72308910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370195);