#7627322

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×#3494	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7274 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,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	#72238225, #72238229, #72238254
maximal normal subgroup of	#72238225, #72238229, #72238254
maximal quotient of	#72263785

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534518);