#6460002

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
prime radical	#23391	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2423 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)(13,14,20,17,19,15,18,22)(16,21,27,24,25,23,26,28);(6,9)(10,12)(13,15,19,14)(16,24,25,28)(17,18,22,20)(21,26,23,27);(1,2,3,4)(13,16,18,26,19,25,20,27)(14,21,22,28,15,23,17,24);(5,7,8,11)(6,10,9,12)(13,17,18,14,19,22,20,15)(16,24,26,21,25,28,27,23);(16,25)(21,23)(24,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307380, #72307394, #72307552
maximal normal subgroup of	#72307380, #72307394, #72307552

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367198);