#7623649

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,9,13)(7,15,17,34)(8,19,22,33)(10,24,21,14)(11,27,29,18)(12,26,31,23)(16,35,25,30)(20,32,36,28);(5,7,9,17)(6,11,13,29)(8,20,22,36)(10,25,21,16)(12,30,31,35)(14,32,24,28)(15,33,34,19)(18,26,27,23);(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,28)(17,29)(19,23)(20,30)(21,24)(22,31)(25,32)(26,33)(27,34)(35,36);(5,8)(6,12)(7,16)(9,22)(10,26)(11,28)(13,31)(14,33)(15,34)(17,25)(18,27)(19,24)(21,23)(29,32);(6,13)(7,17)(12,31)(14,24)(16,25)(18,27)(19,33)(20,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343457, #72343827
maximal normal subgroup of	#72343457, #72343827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530845);