#7630453

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
prime radical	C₂²×#534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

9362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267585, #72267587
maximal normal subgroup of	#72267585, #72267587

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537649);