#6459993

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2446 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)(5,6,8,9)(7,10,11,12)(13,14,17,22,18,19,15,23)(16,21,25,27,26,20,24,28);(6,9)(10,12)(13,15)(14,19)(17,18)(20,27)(21,28)(24,25);(1,2,3,4)(13,16,15,24,18,26,17,25)(14,20,23,27,19,21,22,28);(5,7,8,11)(6,10,9,12)(13,16,17,25,18,26,15,24)(14,21,22,27,19,20,23,28);(14,19)(20,21)(22,23)(27,28)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,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,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216163
maximal normal subgroup of	#72216163
maximal quotient of	#72216022, #72216058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367189);