#7625282

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

8406 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ch

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226739, #72315262, #72315266, …
maximal normal subgroup of	#72226739, #72315262, #72315266, …
maximal quotient of	#72316293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532478);