#6465637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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

3294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,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	#72216157, #72312900, #72313004
maximal normal subgroup of	#72216157, #72312900, #72313004
maximal quotient of	#72216058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372833);