#6467619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	#25787	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2355 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263899, #72263990, #72264026
maximal normal subgroup of	#72263899, #72263990, #72264026
maximal quotient of	#72264943, #72265295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374815);