#6463223

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2797 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)(13,14,20,17)(15,24,26,16)(18,23,19,28)(21,22,27,25);(1,2,3,4)(7,8)(10,12)(13,15,19,25,20,26,18,22)(14,21,23,24,17,27,28,16);(5,7,9,8)(6,10,11,12)(13,16,19,21,20,24,18,27)(14,22,23,15,17,25,28,26);(5,8,9,7)(6,10,11,12)(13,17,20,14)(15,21,26,27)(16,25,24,22)(18,23,19,28);(13,18,20,19)(14,23,17,28)(15,22,26,25)(16,21,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224734, #72239652, #72257004, …
maximal normal subgroup of	#72224734, #72239652, #72257004, …
maximal quotient of	#72224721, #72224726, #72239613

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370419);