#6463169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2830 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)(13,14,19,22)(15,17,21,18)(16,24,27,23)(20,25,28,26);(1,2,3,4)(7,8)(10,12)(13,15,18,14,19,21,17,22)(16,23,26,20,27,24,25,28);(5,7,9,8)(6,10,11,12)(13,16,18,26,19,27,17,25)(14,20,21,24,22,28,15,23);(5,8,9,7)(6,10,11,12);(13,17,19,18)(14,21,22,15)(16,25,27,26)(20,24,28,23)

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^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224791, #72226669, #72282025
maximal normal subgroup of	#72224791, #72226669, #72282025
maximal quotient of	#72224788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370365);