#7654589

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8424 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)(5,6)(7,9)(8,11)(10,12)(13,14,19,17)(15,21,16,20)(18,22,26,25)(23,28,24,27);(1,2,3,4)(6,7)(10,11)(13,14,19,17)(15,20,16,21)(18,22,26,25)(23,27,24,28);(5,7,9,6)(8,11,12,10)(13,15,19,16)(14,20,17,21)(18,23,26,24)(22,27,25,28);(5,8,9,12)(6,10,7,11)(13,16,19,15)(14,21,17,20)(18,24,26,23)(22,28,25,27);(13,17,19,14)(15,21,16,20)(18,22,26,25)(23,27,24,28);(13,18,19,26)(14,22,17,25)(15,23,16,24)(20,27,21,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=hi,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	#72304625, #72304627, #72304639
maximal normal subgroup of	#72304625, #72304627, #72304639

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561785);