#7627235

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8468 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,24)(17,21)(18,23)(19,25)(22,26)(27,28);(6,9)(10,11)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,16,17,25)(14,19,21,24)(15,22,23,28)(18,26,20,27);(13,15)(14,20)(16,22)(17,23)(18,21)(19,27)(24,26)(25,28);(13,14)(15,18)(16,19)(17,21)(20,23)(22,26)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304557, #72304572, #72304604
maximal normal subgroup of	#72304557, #72304572, #72304604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534431);