#7891245

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236836, #72238152, #72238207, …
maximal normal subgroup of	#72236836, #72238152, #72238207, …
maximal quotient of	#72281019, #72281837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7798441);