#7653873

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
prime radical	C₂×#3519	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8364 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)(13,14,19,24)(15,18,23,20)(16,22,17,21)(25,28,26,27);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,26)(18,24)(19,23)(21,28)(22,27);(5,7)(6,9)(8,12)(10,11)(13,16,19,17)(14,21,24,22)(15,25,23,26)(18,27,20,28);(13,17,19,16)(14,22,24,21)(15,26,23,25)(18,28,20,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,25)(17,26)(19,23)(20,24)(21,27)(22,28);(5,6)(7,9)(8,11)(10,12)(13,18)(14,23)(15,24)(16,27)(17,28)(19,20)(21,26)(22,25)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304555, #72304595, #72304614
maximal normal subgroup of	#72304555, #72304595, #72304614

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561069);