#7850809

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

8233 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)(13,14,20,25)(15,21,26,27)(16,23,18,22)(17,19,24,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,21)(16,17)(18,24)(19,23)(20,26)(22,28)(25,27);(5,7)(6,9)(8,12)(10,11)(13,16,20,18)(14,22,25,23)(15,24,26,17)(19,27,28,21);(13,17,20,24)(14,19,25,28)(15,16,26,18)(21,23,27,22);(5,7)(6,10)(8,12)(9,11)(13,18,20,16)(14,23,25,22)(15,24,26,17)(19,27,28,21);(5,6)(7,9)(8,11)(10,12)(13,19)(14,24)(15,23)(16,27)(17,25)(18,21)(20,28)(22,26)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=h,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	#72279324, #72283039, #72284133, …
maximal normal subgroup of	#72279324, #72283039, #72284133, …
maximal quotient of	#72281277, #72281328, #72282310

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7758005);